We can now use the laws of indices and logarithms to solve equations that were previously difficult or impossible to solve.
▸ Solving Index Equations — If bases are the same, equate the exponents. If bases are different, take the logarithm of both sides.
Example: Solve 2^x = 32. Since 32 = 2^5, we have 2^x = 2^5, so x = 5. Solve 5^x = 200 by taking log of both sides: x log 5 = log 200, so x = log 200 / log 5 ≈ 3.292.
▸ Solving Logarithm Equations — Convert the logarithmic equation to exponential form, or use logarithm laws to simplify before converting.
Example: Solve log_3(x) = 4. Convert to exponential form: x = 3^4 = 81. Solve log(x+2) + log(x-1) = 1. Using Product Law: log[(x+2)(x-1)] = 1. Convert to exponential form: (x+2)(x-1) = 10^1 = 10. Solve the quadratic equation x^2 + x - 12 = 0 to get x=3 (x=-4 is rejected).
★ Real-world — Compound Interest:
To find when Ksh 50,000 at 8% p.a. compound interest doubles, we solve 100,000 = 50,000 * (1.08)^t. This simplifies to 2 = (1.08)^t. Using logarithms, t = log 2 / log 1.08 ≈ 9.0 years.
Audio narration — reads aloud, works offline
In your own words, summarise the main idea of this lesson.