Logarithm Calculator

What is a Logarithm?

A logarithm is simply the inverse of an exponent. While an exponent tells you what happens when you raise a base to a power (e.g., $10^2 = 100$), a logarithm asks the reverse question: "To what power must I raise this base to get this number?"

Common Logarithms You Will Encounter

• Common Log (Base 10): Written as $\log(x)$ or $\log_{10}(x)$. This is the default on scientific calculators. For example, $\log_{10}(100) = 2$ because $10^2 = 100$.
• Natural Log (Base e): Written as $\ln(x)$. It uses Euler's number ($e \approx 2.718$) as the base and is vital for continuous growth calculations in calculus.
• Binary Log (Base 2): Written as $\log_2(x)$. Heavily used in computer science. For example, $\log_2(8) = 3$ because $2^3 = 8$.

The Change of Base Formula

Most scientific calculators only have buttons for log (Base 10) and ln (Base e). What happens if your homework asks you to calculate $\log_2(8)$ or $\log_3(81)$?

You must use the Change of Base Formula, which lets you rewrite any logarithm in terms of base 10 or base e:

$\log_b(x) = \frac{\log_{10}(x)}{\log_{10}(b)} = \frac{\ln(x)}{\ln(b)}$

Example: Calculating $\log_2(8)$ on a standard calculator:

1. Rewrite as: $\frac{\log_{10}(8)}{\log_{10}(2)}$
2. Calculate the top: $\log_{10}(8) \approx 0.903$
3. Calculate the bottom: $\log_{10}(2) \approx 0.301$
4. Divide: $\frac{0.903}{0.301} = 3$

Our calculator above automatically applies the change of base formula and shows you the exact steps for any custom base you enter.

Important Logarithm Rules to Remember

When simplifying algebraic expressions, students frequently rely on these core properties:

1. Product Rule: $\log(xy) = \log(x) + \log(y)$
2. Quotient Rule: $\log(\frac{x}{y}) = \log(x) - \log(y)$
3. Power Rule: $\log(x^n) = n \cdot \log(x)$
4. Log of 1: $\log_b(1) = 0$ (because any number to the power of 0 is 1)
5. Log of the Base: $\log_b(b) = 1$ (because $b^1 = b$)