11/14 11. Logarithmic Functions and Equations Free Online FlashCards

Evaluate $\log_5(125)$.

$\log_5(125)=3$, because $5^3=125$.

What are common and natural logarithms?

A common logarithm has base $10$: $\log(x)=\log_{10}(x)$. A natural logarithm has base $e$: $\ln(x)=\log_e(x)$.

State the product property of logarithms.

The product property is $\log_b(MN)=\log_b(M)+\log_b(N)$, for positive $M$ and $N$.

State the quotient property of logarithms.

The quotient property is $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$, for positive $M$ and $N$.

How does the power property rewrite $\log_b(M^p)$?

The power property is $\log_b(M^p)=p\log_b(M)$. The exponent becomes a coefficient.

What is the change-of-base formula?

$\log_b(x)=\frac{\ln(x)}{\ln(b)}$, so $\log_5(17)=\frac{\ln(17)}{\ln(5)}$.

Can $\log_b(x+y)$ be split into two logarithms?

No. Logarithm properties do not separate sums: $\log_b(x+y)\neq\log_b(x)+\log_b(y)$.

What are the domain, range, and asymptote of $y=\log_b(x)$?

For $y=\log_b(x)$, the domain is $(0,\infty)$, the range is all real numbers, and the vertical asymptote is $x=0$.

When is a parent logarithmic graph increasing?

The graph of $y=\log_b(x)$ is increasing when $b>1$ and decreasing when $0<b<1$.

How do $h$ and $k$ affect $a\log_b(x-h)+k$?

For $f(x)=a\log_b(x-h)+k$, the domain is $x>h$, the vertical asymptote is $x=h$, and the range remains all real numbers.

Solve $\log_3(2x-1)=4$.

Convert to exponential form: $2x-1=3^4=81$. Thus $x=41$, and the logarithm argument is positive.

Solve $\log_5(x+1)=\log_5(3x-7)$.

Because the bases match, set the arguments equal: $x+1=3x-7$. Therefore, $x=4$, and both arguments equal $5$.