04 Logarithmic Functions Free Online FlashCards

What restrictions define $\log_b x$?

The base must satisfy $b>0$ and $b\ne1$, while the logarithm's argument must satisfy $x>0$.

Simplify $\log_b(b^x)$.

The identity is $\log_b(b^x)=x$, because logarithms and exponentials with the same base undo each other.

How does the base affect a logarithm's monotonicity?

If $b>1$, $\log_b x$ is increasing; if $0<b<1$, it is decreasing.

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 $N$.

State the power property of logarithms.

The power property is $\log_b(M^r)=r\log_b M$, for positive $M$ and real $r$.

How can you evaluate $\log_b M$ using natural logarithms?

The change-of-base formula is $\log_b M=\frac{\ln M}{\ln b}$, with $M>0$, $b>0$, and $b\ne1$.

Solve $\log_4(2x-1)=3$.

Rewrite in exponential form: $2x-1=4^3=64$. Thus $x=\frac{65}{2}$, which satisfies the domain condition.

Solve $5^x=17$.

Take natural logarithms: $x\ln5=\ln17$. Therefore, $x=\frac{\ln17}{\ln5}\approx1.760$.

When does a logarithmic inequality reverse direction?

For $b>1$, the logarithm is increasing, so the inequality direction is preserved; for $0<b<1$, it is reversed.

How do you solve $A(t)=A_0e^{kt}$ for time?

For $A(t)=A_0e^{kt}$, solve for time with $t=\frac{1}{k}\ln\left(\frac{A}{A_0}\right)$, assuming the ratio is positive.

What is the half-life formula for exponential decay?

For $A(t)=A_0e^{kt}$ with $k<0$, the half-life is $T=-\frac{\ln2}{k}$.