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In this video, we will learn how to convert equations between logarithmic and exponential form.
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We will begin by identifying what an equation looks like in both of these forms.
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A logarithmic function is the inverse or opposite of an exponential function.
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This means that all exponential equations can be written in logarithmic form and vice versa.
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Where 𝑥 is greater than naught, 𝑎 is greater than naught, and 𝑎 is not equal to one, 𝑦 is equal to the log base 𝑎 of 𝑥 is equivalent to 𝑎 to the power of 𝑦 equals 𝑥.
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Any equation written in logarithmic form has an equivalent equation in exponential form.
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We will now look at a question where we need to change an equation in exponential form to one in logarithmic form.
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Express four to the power of negative two equals one sixteenth in its equivalent logarithmic form.
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Our initial equation is written in exponential form.
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This means that it is written 𝑎 to the power of 𝑥 is equal to 𝑦.
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We know that if 𝑎 to the power of 𝑥 is equal to 𝑦, then 𝑥 is equal to log base 𝑎 of 𝑦.
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Any exponential equation has an equivalent logarithmic equation.
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In this question, the value of 𝑎 is four, 𝑥 is equal to negative two, and 𝑦 is equal to one sixteenth.
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We can therefore conclude that negative two is equal to log base four of one sixteenth.
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The equivalent logarithmic form to the exponential equation four to the power of negative two equals one sixteenth is negative two is equal to log base four of one sixteenth.
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In our next question, we’ll convert an equation in logarithmic form to its equivalent exponential form.
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Express log base 20 of 𝑧 equals one-half in its equivalent exponential form.
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We are given an equation in this question in logarithmic form.
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These can be written log base 𝑎 of 𝑦 is equal to 𝑥.
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We know that any equation in logarithmic form will have an equivalent equation in exponential form such that if log base 𝑎 of 𝑦 is equal to 𝑥, then 𝑦 is equal to 𝑎 to the power of 𝑥.
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In this question, the base 𝑎 is equal to 20, 𝑦 is equal to 𝑧, and 𝑥 is equal to one-half.
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The equation can therefore be rewritten as 𝑧 is equal to 20 to the power of a half.
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This is the equivalent exponential form to log base 20 of 𝑧 equals one-half.
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Whilst it is not required in this question, we recall that 𝑥 to the power of a half is the same as the square root of 𝑥.
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This means that 20 to the power of a half is equal to the square root of 20 which, using our laws of radicals or surds, simplifies to two root five.
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𝑧 is equal to two root five.
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As we were just asked to give our answer in exponential form though, the answer is 𝑧 is equal to 20 to the power of a half.
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In our next two questions, we will deal with logarithmic form to the base 10.
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This is known as the common logarithm.
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Express 10 cubed equals 1000 in its equivalent logarithmic form.
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The equation that we are given in this question is written in exponential form of the type 𝑎 to the power of 𝑥 is equal to 𝑦.
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We know that any equation in exponential form has an equivalent logarithmic form.
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If 𝑎 to the power of 𝑥 is equal to 𝑦, then 𝑥 is equal to log base 𝑎 of 𝑦.
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In this question, the base 𝑎 is equal to 10, the exponent 𝑥 is equal to three, and 𝑦 is equal to 1000.
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This means that we can rewrite the equation as three is equal to log base 10 of 1000.
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We recall that log base 10 is called the common logarithm.
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This means that when a logarithm is written without a base, it is assumed to be base 10.
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The log button that can be found on some scientific calculators is log to the base 10.
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When working in base 10, we do not need to write the base.
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Therefore, log of 1000 is equal to three.
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This is the equivalent logarithmic form to 10 cubed or 10 to the power of three is equal to 1000.
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Express log of one million equals six in its equivalent exponential form.
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The equation we are given is written in logarithmic form, which has a general form of log base 𝑎 of 𝑦 is equal to 𝑥.
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We know that any equation in logarithmic form has an equivalent equation in exponential form.
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If log base 𝑎 of 𝑦 is equal to 𝑥, then 𝑦 is equal to 𝑎 to the power of 𝑥.
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We notice in this question that there is no base.
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When any logarithm is shown without a base, it is assumed to be base 10.
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This is known as the common logarithm.
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We can therefore see that 𝑎 is equal to 10, 𝑦 is equal to one million, and 𝑥 is equal to six.
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Rewriting the equation in exponential form, we have one million is equal to 10 to the sixth power or 10 to the power of six.
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We know this answer is correct as 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 is equal to one million.
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When raising 10 to a power, the exponent corresponds to the number of zeros.
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In our final two questions, we will be dealing with the natural logarithm.
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Write the exponential equation 𝑒 to the power of 𝑥 equals five in logarithmic form.
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In order to answer this question, we need to recall the definition of the natural logarithm.
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When 𝑒 to the power of 𝑦 is equal to 𝑥, then base 𝑒 logarithm of 𝑥 is log base 𝑒 of 𝑥 which is written ln of 𝑥 which equals 𝑦.
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The natural logarithm function is the inverse of the exponential function.
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In this question, we are given the exponential equation 𝑒 to the power of 𝑥 is equal to five.
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Rewriting this in logarithmic form, we have 𝑥 is equal to the natural logarithm of five.
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The exponential equation 𝑒 to the power of 𝑥 equals five written in logarithmic form is 𝑥 is equal to ln of five.
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Write the logarithmic equation eight equals ln of 𝑥 in exponential form.
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In this question, we are given the natural logarithm ln of 𝑥.
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We know that the natural logarithm ln of 𝑥 is equal to log of base 𝑒 of 𝑥.
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We also know that ln is the inverse function of the exponential function.
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If 𝑦 is equal to ln of 𝑥, then 𝑒 to the power of 𝑦 equals 𝑥.
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In this question, eight is equal to ln of 𝑥.
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This means that 𝑒 to the power of eight or 𝑒 to the eighth power is equal to 𝑥.
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The logarithmic equation eight equals ln 𝑥 written in exponential form is 𝑒 to the power of eight equals 𝑥.
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We will now summarize the key points from this video.
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We began this video by recalling that a logarithmic function is the inverse of an exponential function.
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This means that every logarithmic equation has an equivalent exponential equation.
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If 𝑥 is equal to log base 𝑎 of 𝑦, then 𝑎 to the power of 𝑥 is equal to 𝑦.
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This allows us to convert from a logarithmic equation to an exponential equation and vice versa.
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We also saw that a logarithm written without a base is assumed to be base 10.
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This is known as the standard logarithm.
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We also saw that the natural logarithm is expressed ln of 𝑥.
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This is the same as log base 𝑒 of 𝑥.
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The natural logarithm function is the inverse of the exponential function.
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This means that if 𝑦 is equal to ln of 𝑥, then 𝑒 to the power of 𝑦 equals 𝑥.