Laws of Logarithms (Edexcel A-Level Mathematics): Revision Notes

Example 1: $5^x = 2.8$

• Problem: $x$ is in the power and needs to be "brought down."

• Solution: Take logs of both sides. $\log(5^x) = \log(2.8)$

• Apply the logarithm rule: $\log(a^b) = b\log(a)$ $x \cdot \log(5) = \log(2.8)$

• Solve for $x$: $x = \frac{\log(2.8)}{\log(5)} \approx 0.640 \quad \text{(3sf)}$

Example 2: $2^{x+3} = 7.2$

$\log(2^{x-3}) = \log(7.2)$

• Apply the logarithm rule: $(x - 3) \cdot \log(2) = \log(7.2)$

• Solve for $x$: $x - 3 = \frac{\log(7.2)}{\log(2)} \approx 2.85 \quad \text{(3sf)}$

$x \approx 2.85 + 3 = 5.85 \quad \text{(3sf)}$

Example 3: $\log(2^{x+3}) = 2\log(3) + \log(5^{4-x})$

• Expand and rearrange: $(x + 3)\log(2) = 2\log(3) + (4-x)\log(5)$

• Group the logarithm terms: $x\log(2) + 3\log(2) = 2\log(3) + 4\log(5) - x\log(5)$

• Collect like terms: $x\log(2) + x\log(5) = 2\log(3) + 4\log(5) - 3\log(2)$

$x(\log(2) + \log(5)) = 2\log(3) + 4\log(5) - 3\log(2)$

• Solve for $x$: $x = \frac{2\log(3) + 4\log(5) - 3\log(2)}{\log(2) + \log(5)}$

Final Simplified Equation:

$x(\log(2) + \log(5)) = 2\log(3) + 4\log(5) - 3\log(2)$

$x = \frac{\log(2^2) + \log(5^4) - \log(2^3)}{\log(2) + \log(5)} \approx 1.73$

Example 1: $\log_{6}(x) = 0.2$

$\Rightarrow 6^{\log_{6}(x)} = 6^{0.2} \quad \text{(Raise } 6 \text{ to both sides)}$

$\Rightarrow x = 6^{0.2} = 1.431\ldots$ $(4sf)$

Example 2: $\log_{12}(4x) = 1/3$

$\Rightarrow 12^{\log_{12}(4x)} = 12^{1/3} \quad \text{(Raise } 12 \text{ to both sides)}$

$ \Rightarrow 4x = 12^{1/3}

$

$\Rightarrow x = \frac{12^{1/3}}{4} = 0.5724\ldots \text{ (Approx.)}$ $(4sf)$

Example 3: $\log(x) + 2 \log(x) = 4$

$ \Rightarrow \log(x )+

\log(x^{2}) = 4$

\Rightarrow$$\log(x^{3}) = 4

$ \Rightarrow 10^{\log(x^{3})} = 10^4 \Rightarrow x^{3} = 10^4

$

$x =10^\frac{{4}}{3} = 21.54 \ (4sf)$

Wrong Method:

$10^{log(x)}+10^{2logx}=10^4$ because

$1+2 = 3 \nRightarrow 10^1+10^2=10^3$

Ensure a single term on each side

1. $\log_{10}\left(\frac{x^2 - 10}{x}\right) \quad \text{(Subtraction rule)}$

2. $\log_{10}\left(\frac{x^2 - 10}{x}\right) = 2 \log_{10} (3)$

$\Rightarrow \log_{10}\left(\frac{x^2 - 10}{x}\right) = \log_{10}(9)$

$\Rightarrow 10^{\log_{10}\left(\frac{x^2 - 10}{x}\right)} = 10^{\log_{10}(9)}$

$\Rightarrow \frac{x^2 - 10}{x} = 9$

$\Rightarrow x^2 - 10 = 9x$

$\Rightarrow x^2 - 9x - 10 = 0$

$\Rightarrow (x - 10)(x + 1) = 0$

$\Rightarrow x = 10 \quad \text{or} \quad x = -1$