Question

Angie is working on solving the exponential equation 23x = 6; however, she is not quite sure where to start. Using complete sentences, describe to Angie how to solve this equation.

Hint: Use the change of base formula: log base b of y equals log y over log b.

Answer 1

x = log₂₃(6) = log(6)/log(23)

Explanation:

You want a description of the solution to the exponential equation 23^x = 6.

Logs

The relevant logarithm relations are ...

`a^x=b\ \longleftrightarrow\ \log_a(b)=x\\\\\log(a^x)=x\log(a)\\\\\log_b(y)=(\log(y))/(\log(b))`

Solution

Angie can use the definition of a logarithm, shown above as the first of the relevant logarithm relations, to rewrite the equation as a log equation:

log₂₃(6) = x

In order to evaluate this expression, Angie can use the change of base formula to write the value of x in terms of logarithms we can find the values of :

x = log₂₃(6) = log(6)/log(23)

This is essentially identical to the solution Angie would reach if she were to take the logarithms of both sides of the original equation, then solve for x.

log(23^x) = log(6) . . . . take logs of both sides

x·log(23) = log(6) . . . . simplify log(23^x)

x = log(6)/log(23) . . . . divide by the coefficient of x

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Additional comment

We prefer the latter approach, as it gets you directly to something you can evaluate. The change of base formula is only needed if you want to write the solution as a base-23 logarithm.

x ≈ 0.571444

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