Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — solving an exponential equation
Solve 32x=50, giving your answer to 3 significant figures.
- 1The unknown is in the exponent, so take logarithms of both sides.Any base works; natural logs are convenient.
- 2ln(32x)=ln50
- 32xln3=ln50The power law brings the exponent down in front.
- 4x=2ln3ln50Divide by everything multiplying x.
- 5x=1.78 (3 s.f.)Check: 33.56≈50 ✓
Example 2 — a logarithmic equation with a rejected root
Solve log2x+log2(x−2)=3.
- 1Combine the two logs into one: log2(x(x−2))=3Adding logs multiplies their arguments.
- 2Undo the logarithm: x(x−2)=23=8A log equals 3 means the base raised to 3.
- 3x2−2x−8=0Rearrange to zero.
- 4(x−4)(x+2)=0, so x=4 or x=−2
- 5Reject x=−2: it would require log2(−2), which does not exist.This rejection carries a mark.
- 6x=4. Check: log24+log22=2+1=3 ✓Always test in the original equation.
Example 3 — reducing a power law to linear form
Experimental data is believed to follow y=axn. Explain how to find a and n graphically.
- 1Take logarithms of both sides: logy=log(axn)
- 2logy=loga+nlogxThe product becomes a sum; the power comes down.
- 3Compare with Y=mX+c where Y=logy and X=logx.Both axes are logarithmic for a power law.
- 4The gradient of the line is n.Read it straight off.
- 5The intercept is loga, so a=10intercept.The intercept is not a itself — this is where most marks are lost.