The logarithm laws are consequences of exponent laws.
For example, $MN=a^p\cdot a^q=a^{p+q}$ leads to $\log_a(MN)=p+q=\log_a M+\log_a N$.
Solve Without a Calculator by Recognizing Powers
If you can express the number as an exact power of the base, you can solve instantly.
Example
$\log_5 625 = x$. Since $625=5^4$, you get $x=4$.
$\log_4 0.0625 = x$. Since $0.0625=\dfrac{1}{16}=4^{-2}$, you get $x=-2$.
$\log_2 0.125 = x$. Since $0.125=\dfrac{1}{8}=2^{-3}$, you get $x=-3$.
$\log_3 1=x$. Since $1=3^0$, you get $x=0$.
Examples with Fractional Exponents
Fractional exponents usually come from roots.
Example
$\log_5 \sqrt{5}=x$:
Since $\sqrt{5}=5^{1/2}$, you get $x=\dfrac{1}{2}$.
$\log_3\left(3\sqrt{3}\right)=x$:
Rewrite: $3\sqrt{3}=3\cdot 3^{1/2}=3^{3/2}$, so $x=\dfrac{3}{2}$.
Tip
When you see a square root, immediately think "power $1/2$".
When you see a cube root, think "power $1/3$".
This often turns a messy log into a one-line solution.
Change of Base Makes Any Logarithm Calculator-Friendly
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 $\ln$).
Example question
Solve $2^x=50$.
Solution
Rewrite: $$x=\log_2 50$$
Use change of base with $c=10$: $$x=\frac{\log 50}{\log 2}\approx 5.64 \text{ (3 s.f.)}$$
This method is also how you evaluate expressions such as $\log8 5$ or $\log6 121$ to a requested number of decimal places.
Use Logarithms to Solve Exponential Equations
Many equations have $x$ in the exponent, for example $2^x=50$ or $6^{2x+1}=20$.
Taking a logarithm converts "exponent problems" into "algebra problems".
General Method (One Unknown Exponent)
To solve $a^{f(x)}=b$:
Rewrite as $\log_a b=f(x)$, or take logs of both sides.
Solve the resulting linear equation in $x$.
Exam technique
If the base is not 10 or $e$, a calculator usually cannot compute $\log_a b$ directly.
Use the change of base formula to convert it to $\log$ or $\ln$.
Example question
Solve $6^{2x+1}=20$.
Solution
Rewrite in logarithmic form: $$\log_6 20 = 2x+1$$
Use the change of base formula to evaluate $\log_6 20$ on a calculator, then solve: $$x=\frac{\log_6 20-1}{2} \approx 0.34$$ (rounded to 2 decimal places).
Example question
Solve $3^x=40$.
Solution
Convert to logarithms: $$x=\log_3 40$$
Then compute with a calculator (using change of base) and round to the required accuracy: $$x \approx 3.36$$
Common Mistake
Do not round too early.
Keep calculator values in full (or at least 4 to 6 significant figures) until the final step, then round to the accuracy requested.
Log and Exponential Graphs Help You See What Is Happening
Logarithm and exponential functions with the same base are inverse functions, so their graphs are mirror images in the line $y=x$.
Key features to connect to solving equations:
Solving $2^x=8$ means finding where the exponential graph hits $y=8$.
Solving $\log_2 x=3$ means finding the $x$-value where the log graph hits $y=3$.
These solutions match because the functions are inverses.
Analogy
Think of $y=2^x$ as a "machine" that takes an input $x$ and outputs a power of 2.
The logarithm $y=\log_2 x$ is the "undo machine" that takes a power of 2 and returns the exponent you started with.
Solve Exponential Equations by Writing Both Sides with the Same Base
Sometimes you can avoid calculators entirely by expressing both sides as powers of the same base, then equating exponents.
Strategy:
If you can rewrite both sides as $k^{\text{(something)}}$, then: $$k^{A(x)}=k^{B(x)} \Rightarrow A(x)=B(x)$$
Example question
Solve $2^{x+1}=4^{2x}$.
Solution
Rewrite $4^{2x}$ using base 2: $$4^{2x}=(2^2)^{2x}=2^{4x}$$
So: $$2^{x+1}=2^{4x} \Rightarrow x+1=4x \Rightarrow x=\frac{1}{3}$$
Example question
Solve $\left(\frac{1}{3}\right)^{x+2}=9^{2x-2}$.
Solution
Rewrite both sides in base 3: $$\left(\frac{1}{3}\right)^{x+2}=3^{-(x+2)}$$ $$9^{2x-2}=(3^2)^{2x-2}=3^{4x-4}$$