Exponential and Logarithmic Functions
5.6 Logarithmic Properties
Core Log Rules
These properties turn products and powers into simpler sums and multiples.
Theorem: Logarithm properties
For $M,N>0$:
- Product rule: $\log_b(MN)=\log_b M+\log_b N$
- Quotient rule: $\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N$
- Power rule: $\log_b(M^k)=k\log_b M$
Theorem: Change of base
For $b,k>0$, $b\ne1$, $k\ne1$:
$$\log_b M=\frac{\log_k M}{\log_k b}. $$
Example: Expand a logarithm
Expand fully:
$$\log_3\left(\frac{x^2\sqrt{y}}{z^4}\right).$$
Show Solution
Use quotient, product, and power rules:
$$ \log_3(x^2\sqrt{y})-\log_3(z^4) =\log_3(x^2)+\log_3(y^{1/2})-\log_3(z^4) =2\log_3 x+\frac12\log_3 y-4\log_3 z. $$
MyOpenMath: Practice: expanding logarithms (placeholder)
Combine Logarithmic Expressions
Combining is the reverse process of expansion.
Example: Condense to one logarithm
Condense:
$$3\log x-2\log y+\frac12\log z.$$
Show Solution
Move coefficients as exponents:
$$\log(x^3)-\log(y^2)+\log(z^{1/2}).$$
Combine product and quotient:
$$\log\left(\frac{x^3\sqrt{z}}{y^2}\right).$$