Question

How do I solve for x here? Use the properties of logarithms to find a value for x. Assume a,b, and M are constants.

Answer 1

Yes, you're right! The first step is rewriting the equation as

$\ln(a) + \ln(b^x) = M$

Subtract $\ln(a)$ from both sides:

$\ln(b^x) = M-\ln(a)$

Use the property $\ln(a^b) = b\ln(a)$ to rewrite the equation as

$x\ln(b) = M-\ln(a)$

Divide both sides by $\ln(b)$

$x = \dfrac{M-\ln(a)}{\ln(b)}$

Alternative strategy:

Consider both sides as exponents of e:

$e^{\ln(ab^x)} = e^M$

Use $e^{\ln(x)} = x$ to write

$ab^x = e^M$

Divide both sides by a:

$b^x = \dfrac{e^M}{a}$

Consider the logarithm base b of both sides:

$x = \log_b\left(\dfrac{e^M}{a}\right)$

The two numbers are the same: you can check it using the rule for changing the base of logarithms