The 5 logarithm properties to know 100%

The 5 formulas to memorize

For $a, b > 0$ and $n \in \mathbb{R}$:

1. $\ln(a \times b) = \ln a + \ln b$ (product → sum)
2. $\ln\!\left(\dfrac{a}{b}\right) = \ln a - \ln b$ (quotient → difference)
3. $\ln(a^n) = n \ln a$ (power → coefficient)
4. $\ln(\sqrt{a}) = \dfrac{1}{2} \ln a$
5. $\ln(1) = 0$ and $\ln(e) = 1$

The key move

Faced with $\ln(\text{something big})$, try to break down the inside into a product, quotient or power, then apply the formulas.

An example

$\ln\!\left(\dfrac{e^3 \times 5}{2}\right) = \ln(e^3) + \ln 5 - \ln 2 = 3 + \ln 5 - \ln 2$.

Solve $\ln x = 2$

Apply the exponential to both sides: $x = e^2 \approx 7.39$.

Rule: $\ln$ and $\exp$ are inverses. $\ln(e^x) = x$, $e^{\ln x} = x$ (for $x > 0$).

Watch out!

$\ln(a + b) \ne \ln a + \ln b$. Log does NOT turn sums into products, only products into sums. A very common trap.

?Your turn

What is ln(e⁵ / √e)?

A.5B.4.5C.10/√eD.e^4.5

Check