Question

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

Answer 1

`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
`x` 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))}`