Logarithms: Concepts, Formulas, and Clear Examples

Logs invert exponentials: if $b^x = N$, then $\log_b N = x$. MCQs test change of base, log laws, and solving $\log$ equations with domain $N>0$, $b>0$, $b\neq1$.

Memorize $\log_{10}$ and $\ln$ conversions only when stem gives approximate values; exact laws beat calculator.

Combine logs before expanding—fewer steps, fewer sign errors.

Definition and domain

Definition

$$b^x = N \;\Leftrightarrow\; \log_b N = x$$

Domain

$$N > 0,\ b > 0,\ b \neq 1$$

Basic laws

Product

$$\log_b(MN) = \log_b M + \log_b N$$

Quotient

$$\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$$

Power

$$\log_b(M^k) = k\log_b M$$

Change of base

Change of base

$$\log_b N = \frac{\log_k N}{\log_k b} = \frac{\ln N}{\ln b}$$

Example — Evaluate log_2 8

$2^x=8$ → $x=3$. Or $\frac{\ln 8}{\ln 2}=3$.

Common values

• $\log_{10} 10=1$, $\log_b 1=0$, $\log_b b=1$.
• $\log_b(b^k)=k$.

Solving log equations

Equate arguments

$$\log_b M = \log_b N \;\Rightarrow\; M = N$$

• Combine to single log each side, then equate arguments if $\log_b$ same base.
• Always check solutions in original domain.

Exponential form tricks

Switch

$$a^{\log_a x} = x$$

Step-by-step method for exam questions

1. Confirm base and argument positivity.
2. Compress/expand using log laws.
3. Change base if bases differ.
4. Solve linear equation in $\log x$ or equate arguments.
5. Reject extraneous (negative/zero inside log).
6. Pick option.

Formula cheat sheet

Frequently asked questions

Can log be negative?

Yes, for $0<N<1$ with base $>1$.

log(-5)?

Undefined in reals—discard such roots.

ln vs log?

ln is $\log_e$; stem usually specifies base 10 or e.

Forgot to check domain?

Classic trap—always substitute back.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.