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. ln(a*b^(x)) = M The answer in the back of the textbook is x=(…

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asked May 27, 2019 138k views

1 Answer

Wombleton · Answer 1 · 2019-06-03T08:13:04+0000

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
$\ln(b)$

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

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 = (e^M)/(a)

Consider the logarithm base b of both sides:

$x = \log_b\left((e^M)/(a)\right)$

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