Simplifying an Expression with Log Laws

The log laws let you combine a string of logarithms into one: additions become a product, subtractions a quotient. Simplify to a single log first, then evaluate the argument.

Given that all logarithms are to base $10$, find the exact value of $\lg 8 + \lg 5 - \lg 4$. [3]

The working

Step 1, combine into a single logarithm. Additions multiply the arguments, the subtraction divides: $\lg 8 + \lg 5 - \lg 4 = \lg\!\left(\frac{8 \times 5}{4}\right) \quad \text{(M1)}$

Step 2, simplify the argument: $= \lg\!\left(\frac{40}{4}\right) = \lg 10 \quad \text{(M1)}$

Step 3, evaluate. Since the base is $10$, $\lg 10 = 1$: $= 1 \quad \text{(A1)}$

Where the marks are won and lost

• Additions multiply, subtractions divide: $\lg 8 + \lg 5 - \lg 4 = \lg\frac{8 \times 5}{4}$. Mixing these up (e.g. adding the arguments) is the classic error.
• Simplify the argument to $\frac{40}{4} = 10$ before taking the log’s value.
• $\lg 10 = 1$ because the base is $10$ and $10^1 = 10$. (If it were $\ln 10$, the answer would be $\approx 2.30$.)

Common mistakes

• Adding the arguments ($8 + 5$) instead of multiplying them.
• Writing $\lg 10 = 10$ or $0$ instead of $1$.
• Confusing $\lg$ (base 10) with $\ln$ (base $e$).

Full method: Laws of Logarithms notes. Topic home: Logs & Exponentials pillar.