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Find the value of log base seven of 32 plus log base seven of eight divided by log base seven of 10 minus log base seven of five without using a calculator.
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Letโs recall some of the laws of logarithms.
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We know that when adding logarithms whose base is equal, we simply multiply the argument.
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So log base ๐ of ๐ฅ one plus log base ๐ of ๐ฅ two is log base ๐ of ๐ฅ one times ๐ฅ two.
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We have a similar rule for subtracting, but this time we divide the arguments.
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And so letโs use these rules to evaluate the numerator and denominator of our fraction.
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Log base seven of 32 plus log base seven of eight is the same as log base seven of 32 times eight, but 32 times eight is 256.
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So our numerator becomes log base seven of 256.
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Then our denominator is log base seven of 10 divided by five, which is log base seven of two.
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And so weโve simplified a little bit, and our fraction becomes log base seven of 256 divided by log base seven of two.
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Now, we need to be really careful here.
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A common mistake is to think that because when we divide the arguments we subtract the two logarithms, we can simply subtract these values.
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Remember, thatโs not actually what our log laws say.
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Instead, weโre going to apply the change of base formula, so called because it literally allows us to change the base that weโre working with.
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For this to work, we need to have a fraction made up of two logarithms whose base is the same.
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So log base ๐ of ๐ฅ one divided by log base ๐ of ๐ฅ two is then log base ๐ฅ two of ๐ฅ one.
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Comparing this general form with our fraction, we find the base ๐ is equal to seven.
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๐ฅ one is the argument of the logarithm on the top of our fraction, so itโs 256.
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And ๐ฅ sub two is the argument of the logarithm on our denominator, so itโs two.
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This means that we can now write log base seven of 256 over log base seven of two as log base two of 256.
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Weโre still not finished though.
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We have fully simplified it, but we need to evaluate this.
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And so letโs recall the definition of a logarithm.
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If we say log base ๐ of ๐ฆ equals ๐ฅ, we can equivalently say that ๐ to the power of ๐ฅ must be equal to ๐ฆ.
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And so, here, since our base is two, weโre asking, what exponent of two gives us a value of 256?
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Well, two to the eighth power is 256.
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And this then means that log base two of 256 must be eight.
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Log base seven of 32 plus log base seven of eight all divided by log base seven of 10 minus log base seven of five is eight.