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Find the value of 𝑥 for which log base 𝑥 of 243 is equal to negative five.

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In order to answer this question, we recall that if log base 𝑎 of 𝑏 is equal to 𝑐, then 𝑎 to the power of 𝑐 is equal to 𝑏.

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In this question, we need to calculate the value of 𝑎.

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We know that 𝑏 is 243 and 𝑐 is equal to negative five.

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This means that log base 𝑥 of 243 equals negative five can be rewritten as 𝑥 to the power of negative five is equal to 243.

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We know that when dealing with negative exponents or indices, 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛.

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This means that one over 𝑥 to the fifth power is equal to 243.

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We can multiply through by 𝑥 to the fifth power, giving us one is equal to 243 multiplied by 𝑥 to the fifth power.

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Dividing both sides of this equation by 243 gives us 𝑥 to the fifth power is equal to one over 243.

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Finally, we can calculate the value of 𝑥 by taking the fifth root of one over 243.

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When taking the root of a fraction, we simply take the root of the numerator and the root of the denominator separately.

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The fifth root of one is equal to one and the fifth root of 243 is equal to three as three to the fifth power is 243.

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The value of 𝑥 for which log to the base 𝑥 of 243 equals negative five is one-third.