Video: Finding the Solution Set of Exponential Equations Using Logarithms

Find the solution set to the equation 15^(𝑥) = 64. Give your answer to three decimal places.

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Video Transcript

Find the solution set to the equation 15 to the power of 𝑥 equals 64. Give your answer to three decimal places.

To work out the value of any exponent in an equation of the form 𝑎 to the power of 𝑏 equals 𝑐, we need to know the link between exponents and logarithms. If 𝑎 to the power of 𝑏 equals 𝑐, then 𝑏 is equal to log 𝑐 to the base 𝑎. In this question, 𝑎 is equal to 15, 𝑏 is equal to 𝑥, and 𝑐 is equal to 64. This means that 𝑥 is equal to log 64 to the base 15. Typing this into the calculator gives us 1.535748 and so on.

We are asked to give our answer to three decimal places. This means that the seven is the deciding number. If the deciding number is five or greater, as in this case, we round up. Therefore, 𝑥 is equal to 1.536. As we are asked to give the solution set, we need to write our answer using set notation as shown. We could check this answer by substituting our value back into the original equation.

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