Solve each log equation below. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places.
(1) -2 log_4 (x) = log_4 (9)
(2) ln (x + 1) - ln x = 2
1.)
$\displaystyle -2 \log _4 (x) = \log_4 (9)$
A general rule is: $\displaystyle a \log (x) = \log (x^a)$
Hence: $\displaystyle \log_4(x^{-2}) = \log_4(9)$
$\displaystyle \log_4\left(\frac{1}{x^2}\right) = \log_4(9)$
As both sides are to the same logarithm base, we can equate the changing variable hence:
$\displaystyle \frac{1}{x^2} = 9$
Now solve for $\displaystyle x$
2.)
$\displaystyle \ln (x+1) - \ln x = 2$
A general rule is: $\displaystyle \ln (a) - \ln (b) = \ln \left(\frac{a}{b}\right)$
Hence: $\displaystyle \ln \left(\frac{x+1}{x}\right) = 2$
A general rule is: $\displaystyle e^{\ln(ax)} = ax$
Hence: $\displaystyle e^{\ln \left(\frac{x+1}{x}\right)} = e^{2}$
$\displaystyle \frac{x+1}{x} = e^{2}$
Now solve for $\displaystyle x$ using algebra skills.