A forensic scientist is analysing a blood sample. The concentration of a drug in the bloodstream halves every 4 hours. To find when the concentration falls below a legal threshold, the scientist must solve $C_0 \cdot (\frac{1}{2})^{t/4} < C_{legal}$. Without logarithms, this equation is trapped — the unknown is in the exponent. The laws of logarithms are the keys that unlock exponential equations.
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Without a calculator, which is larger: $\log_{10}(100) + \log_{10}(1000)$ or $\log_{10}(100) \times \log_{10}(1000)$?
Make a prediction before reading on.
Core Content
Every logarithm law corresponds directly to an exponent law. This is not coincidence — it is the defining property of logarithms.
If $M = a^x$ and $N = a^y$, then $MN = a^x \cdot a^y = a^{x+y}$. Taking $\log_a$ of both sides: $\log_a(MN) = x + y = \log_a(M) + \log_a(N)$.
Key idea: Logs turn multiplication into addition. This is why slide rules worked — adding lengths (logs) is easier than multiplying numbers.
Similarly, $\frac{M}{N} = \frac{a^x}{a^y} = a^{x-y}$, so $\log_a(\frac{M}{N}) = x - y = \log_a(M) - \log_a(N)$.
If $M = a^x$, then $M^n = (a^x)^n = a^{xn}$, so $\log_a(M^n) = xn = n \log_a(M)$.
The most useful law. The power law is what lets us bring exponents down from logarithmic expressions — the key step in solving exponential equations.
Two crucial identities follow directly from the definition of logarithms:
Also useful:
These identities are the foundation for solving equations like $2^x = 10$: take $\log_2$ of both sides to get $x = \log_2(10)$.
Simplify: $\log_2(8) + \log_2(4) - \log_2(2)$.
A single numerical value.
log_2(8) = log_2(2^3) = 3
log_2(4) = log_2(2^2) = 2
log_2(2) = 1
Sum: 3 + 2 - 1 = 4
$\log_2(8) + \log_2(4) - \log_2(2) = 4$.
Simplify $\log_5(125) + 2\log_5(25) - \log_5(5)$.
Answer:
$\log_5(125) = 3$, $\log_5(25) = 2$, $\log_5(5) = 1$. Expression = $3 + 2(2) - 1 = 3 + 4 - 1 = 6$.
Expanding: Breaking a single log into a sum or difference of simpler logs.
Example: log_2(x^3 y / z) = 3 log_2(x) + log_2(y) - log_2(z)
Condensing: Combining a sum or difference of logs into a single log.
Example: 2 ln(x) + ln(y) - 3 ln(z) = ln(x^2 y / z^3)
When condensing, always check that the final expression is defined (all arguments positive).
$\log_a(MN) = \log_a(M) + \log_a(N)$
$\log_a(M/N) = \log_a(M) - \log_a(N)$
$\log_a(M^n) = n\log_a(M)$
$\log_a(a^x) = x$
$a^{\log_a(x)} = x$
$\log_{10}(100) = 2$ and $\log_{10}(1000) = 3$. So $\log_{10}(100) + \log_{10}(1000) = 2 + 3 = 5$, while $\log_{10}(100) \times \log_{10}(1000) = 2 \times 3 = 6$. The product is larger! Many students guess the sum because of the product law — but that law applies to the arguments of the logs, not to the logs themselves.
5 random questions from a replayable lesson bank — feedback shown immediately
Short Answer
8. Simplify $\log_2(32) + \log_2(8) - \log_2(4)$, showing each step. 3 MARKS
9. Expand $\ln(\frac{x^3 \sqrt{y}}{z^2})$ completely. 3 MARKS
10. A drug has concentration $C = C_0 \cdot 2^{-t/3}$ in the bloodstream, where $t$ is in hours. The legal driving limit is $C = \frac{C_0}{16}$. Use logarithms to find how many hours must pass before it is legal to drive. Explain why logarithms are essential for this calculation. 3 MARKS
1. $\log_3(27) + \log_3(9) = 3 + 2 = 5$.
2. $\log_{10}(1000) - \log_{10}(10) = 3 - 1 = 2$.
3. $2\log_2(8) + \log_2(4) = 2(3) + 2 = 8$.
4. $\ln(x^2 y^3) = 2\ln(x) + 3\ln(y)$.
5. $3\ln(x) - 2\ln(y) + \ln(z) = \ln(\frac{x^3 z}{y^2})$.
1. LHS = $\log_{10}(1100) \approx 3.04$. RHS = $2 + 3 = 5$. Not equal. The product law applies to multiplication inside the log, not multiplication of logs.
2. $C/C_0 = (1/2)^{t/4}$. Take $\log_{1/2}$: $\log_{1/2}(C/C_0) = t/4$, so $t = 4\log_{1/2}(C/C_0)$.
3. No. Counterexample: $\log_{10}(100) \cdot \log_{10}(1000) = 2 \times 3 = 6$, but $\log_{10}(100{,}000) = 5$.
Q8 (3 marks): $\log_2(32) = 5$ [1], $\log_2(8) = 3$ [0.5], $\log_2(4) = 2$ [0.5]. Expression = $5 + 3 - 2 = 6$ [1].
Q9 (3 marks): $= \ln(x^3) + \ln(\sqrt{y}) - \ln(z^2)$ [0.5] $= 3\ln(x) + \frac{1}{2}\ln(y) - 2\ln(z)$ [2.5]. Accept equivalent forms with powers expressed as fractions or decimals.
Q10 (3 marks): $\frac{C_0}{16} = C_0 \cdot 2^{-t/3}$ [0.5]. Divide: $\frac{1}{16} = 2^{-t/3}$ [0.5]. Since $\frac{1}{16} = 2^{-4}$, we have $-4 = -t/3$ [0.5], so $t = 12$ hours [0.5]. Logarithms are essential because the unknown $t$ appears in the exponent [0.5]. Without logarithms, we cannot algebraically isolate a variable in the exponent [0.5].
Climb platforms using your knowledge of log laws, simplification, and expansion. Pool: lesson 7.
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