Log Equations

**magentarita** · August 9th 2008, 06:41 AM

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

**Simplicity** · August 9th 2008, 07:05 AM

Originally Posted by magentarita

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.)
$-2 \log _4 (x) = \log_4 (9)$
A general rule is: $a \log (x) = \log (x^a)$
Hence: $\log_4(x^{-2}) = \log_4(9)$
$\log_4\left(\frac{1}{x^2}\right) = \log_4(9)$
As both sides are to the same logarithm base, we can equate the changing variable hence:
$\frac{1}{x^2} = 9$
Now solve for $x$

2.)
$\ln (x+1) - \ln x = 2$
A general rule is: $\ln (a) - \ln (b) = \ln \left(\frac{a}{b}\right)$
Hence: $\ln \left(\frac{x+1}{x}\right) = 2$
A general rule is: $e^{\ln(ax)} = ax$
Hence: $e^{\ln \left(\frac{x+1}{x}\right)} = e^{2}$
$\frac{x+1}{x} = e^{2}$
Now solve for $x$ using algebra skills.

**magentarita** · August 9th 2008, 01:42 PM

Great reply as always. Where have you been, Air?

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