Logarithmic functions and indices (OLevel's Pure Math)

1. The Essentials

Before tackling complex problems, remember the core relationship between indices and logs:

$$a^x=y\iff x={\log }_{a}y$$

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2. Worked Example: Solving Exponential Equations

Question: Solve the equation $3^{2x+1}=5^x$, giving your answer to 3 significant figures.

Step-by-Step Solution:

1. Take the log of both sides:
   $$\log \left(3^{2x+1}\right)=\log \left(5^x\right)$$
2. Apply the Power Law:
   $$(2x+1)\log 3=x\log 5$$
3. Expand the brackets:
   $$2x\log 3+\log 3=x\log 5$$
4. Rearrange to isolate $x$:
   $$2x\log 3-x\log 5=-\log 3$$ $$x(2\log 3-\log 5)=-\log 3$$
5. Calculate the final value:
   $$x=\frac{-\log 3}{2\log 3-\log 5}\approx -1.86$$

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3. Worked Example: Hidden Quadratics

Question: Solve $e^{2x}-5e^x+4=0$.

Step-by-Step Solution:

• Substitute: Let $u=e^x$. The equation becomes: $$u^2-5u+4=0$$
• Factorize: $$(u-4)(u-1)=0$$ So, $u=4$ or $u=1$.
• Solve for $x$:
  If $e^x=4\Rightarrow x=\ln 4\approx 1.39$
  If $e^x=1\Rightarrow x=\ln 1=0$

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4. Linearization of Power Functions

When you have a relationship like $y=a{x}^n$, plotting $\log y$ against $\log x$ results in a straight line.

Step-by-Step Transformation:

$$y=a{x}^n$$ $$\log y=\log \left(a{x}^n\right)$$ $$\log y=\log a+n\log x$$

This matches $Y=C+mX$, where the gradient is $n$ and the vertical intercept is $\log a$.