Sometimes you are given substitutions like $a=\ln x$, $b=\ln(x-1)$, $c=\ln 3$, and you must rewrite more complex expressions using $a,b,c$.
Example question
Given $a=\ln x$ and $b=\ln(x-1)$, express $\ln\left(\frac{x}{x-1}\right)$ in terms of $a$ and $b$.
Solution
Use the quotient rule: $$\ln\left(\frac{x}{x-1}\right)=\ln x-\ln(x-1)=a-b$$
Tip
When you see a square root, rewrite it as a power first: $\sqrt{A}=A^{1/2}$.
Then the power rule applies immediately.
Condensing Multiple Logs into One
To combine logs into a single log:
Use the power rule to move coefficients into exponents.
Turn sums into products (product rule reversed).
Turn differences into quotients (quotient rule reversed).
Example question
Express $3\ln x + 2\ln(2x+1)$ as a single logarithm.
Solution
Power rule: $$3\ln x=\ln(x^3), \qquad 2\ln(2x+1)=\ln\big((2x+1)^2\big)$$
Product rule: $$3\ln x + 2\ln(2x+1)=\ln\left(x^3(2x+1)^2\right)$$
Common Mistake
If you expand $(2x+1)^2$ to $4x^2+4x+1$, do not forget the factor $x^3$ outside.
Many algebra errors happen after the log step, not during it.
Change of Base and Using Log Tables
Definition
Change of base formula
For $a>0$, $a\neq 1$, $b>0$, and any convenient base $c>0$, $c\neq 1$: $$\log_a b=\frac{\log_c b}{\log_c a}$$ Common choices are $c=10$ (using $\log$) or $c=e$ (using $\ln$).