Question Video: Finding the Logarithmic Equivalence to an Exponential Equation Mathematics

Fill in the blank: If 𝑎^(𝑥) = 𝑏, then 𝑥 = _.

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Video Transcript

Fill in the blank: If 𝑎 to the power of 𝑥 equals 𝑏, then 𝑥 equals what.

We begin by recalling that a logarithmic function is the inverse of an exponential function. If we consider the equation two to the power of 𝑥 is equal to eight, which is written in the form 𝑎 to the power of 𝑥 equals 𝑏, then this can also be written as 𝑥 is equal to log base two of eight. To work out the value of 𝑥, we’ll ask ourselves a base of two raised to what power equals eight. We know that two cubed equals eight. Therefore, 𝑥 is equal to three.

This means that log base two of eight is also equal to three. In terms of the general rule, if 𝑎 to the power of 𝑥 equals 𝑏, then 𝑥 is equal to log base 𝑎 of 𝑏. It is important to note that this only holds if 𝑎 and 𝑏 are greater than zero and 𝑎 is not equal to one.

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