Question

1. use logarithms to solve the exponential equation.

$e^{2x}=6$
the solution is $x = \square$.
(type an integer or decimal rounded to three decimal places as needed.)

1. solve for $x$.

$\log_{6}(x + 8)+\log_{6}(x - 8)=2$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. the solution(s) is/are $x = \square$.
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
\\(\bigcirc\\) b. there is no solution.

Explanation:

Problem 10

Step1: Take natural log of both sides

$\ln(e^{2x}) = \ln(6)$

Step2: Simplify left side using $\ln(e^a)=a$

$2x = \ln(6)$

Step3: Solve for x, divide by 2

$x = \frac{\ln(6)}{2}$

Step4: Calculate decimal value

$x \approx \frac{1.791759}{2} \approx 0.896$

Step1: Combine logs using product rule

$\log_6[(x+8)(x-8)] = 2$

Step2: Rewrite in exponential form

$(x+8)(x-8) = 6^2$

Step3: Expand and simplify equation

$x^2 - 64 = 36$
$x^2 = 100$

Step4: Solve for x, check domain

$x = \pm 10$; $x=-10$ makes $\log_6(x-8)$ undefined, so discard.

Answer:

$0.896$

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Problem 11