I need help with 2, i don’t know what to do

Question

asked May 24, 2024 158k views

1 Answer

Jason Armstrong · Answer 1 · 2024-05-30T18:39:58+0000

Answer:

$x=(3\ln(2))/(\ln(7))$

Explanation:

First, we can take the natural log of both sides:

$\ln(14^(6x)) = \ln(64^(x+3))$

And we can use the log exponent rule to simplify ...
$\ln(a^b) = b\cdot \ln(a)$ :

$6x\cdot \ln(14) = (x+3)\cdot \ln(64)$

The log terms can be reduced further by prime factoring the numbers within the log functions:

$6x\cdot \ln(2\cdot7) = (x+3)\cdot \ln(2^6)$

and applying the exponent rule once again, as well as the product rule ...
$\ln(a\cdot b) = \ln(a) + \ln(b)$ :

$6x\cdot [\:\ln(2)+\ln(7)\,] = (x+3)\cdot 6 \ln(2)$

Now, we can solve for x by isolating all terms with x in them:

$x\cdot [\:\ln(2)+\ln(7)\,] = x\ln(2)+3\ln(2)$

↓ subtracting
$x\ln(2)$ from both sides

$x\cdot [\:\ln(2)+\ln(7)\,] - x\ln(2)=3\ln(2)$

↓ factoring an
out of both terms on the left side

$x(\ln(2)+\ln(7)-\ln(2))=3\ln(2)$

↓ executing the subtraction ...
$\ln(2)-\ln(2)=0$

$x\ln(7)=3\ln(2)$

↓ dividing both sides by
$\ln(7)$

$\boxed{x=(3\ln(2))/(\ln(7))}$