Logarithm rules — product, quotient, power, change of base with applications

Question

Simplify: $\log_2 32 + \log_3 81 - \log_5 125$ . Also, if $\log_{10} 2 = 0.3010$ , find $\log_{10} 50$ .

Solution — Step by Step

$\log_2 32 = \log_2 2^5 = 5$ (since $2^5 = 32$ )

$\log_3 81 = \log_3 3^4 = 4$ (since $3^4 = 81$ )

$\log_5 125 = \log_5 5^3 = 3$ (since $5^3 = 125$ )

\log_2 32 + \log_3 81 - \log_5 125 = 5 + 4 - 3 = \mathbf{6}

\log_{10} 50 = \log_{10} \frac{100}{2} = \log_{10} 100 - \log_{10} 2 = 2 - 0.3010 = \mathbf{1.6990}

We used the quotient rule: $\log(a/b) = \log a - \log b$ .

Why This Works

Logarithms convert multiplication into addition — that is their core power. All log rules follow from the definition: $\log_a x = n$ means $a^n = x$ .

graph TD
    A["Logarithm Rules"] --> B["Product: log ab = log a + log b"]
    A --> C["Quotient: log a/b = log a - log b"]
    A --> D["Power: log aⁿ = n log a"]
    A --> E["Change of base: log_a b = log_c b / log_c a"]
    A --> F["Special values"]
    F --> G["log_a a = 1"]
    F --> H["log_a 1 = 0"]
    A --> I["Which rule to apply?"]
    I --> J["See multiplication? Use product rule"]
    I --> K["See division? Use quotient rule"]
    I --> L["See exponent? Use power rule"]
    I --> M["Different bases? Use change of base"]

JEE Main frequently tests logarithm properties in the context of equations. A favourite pattern: $\log_2(x+1) + \log_2(x-1) = 3$ . Using the product rule: $\log_2[(x+1)(x-1)] = 3 \implies x^2 - 1 = 8 \implies x = 3$ (rejecting $x = -3$ since we need both arguments to be positive). Always check the domain.

Alternative Method

The change of base formula is the most versatile tool: $\log_a b = \frac{\ln b}{\ln a} = \frac{\log_{10} b}{\log_{10} a}$

This lets you convert any log to natural or common log, which your calculator can handle. For competitive exams (no calculator), use it to convert everything to the same base before simplifying.

Useful identity: $\log_a b \times \log_b c = \log_a c$ (chain rule for logs). This telescoping property is tested in JEE often.

Common Mistake

Writing $\log(a + b) = \log a + \log b$ . This is WRONG. There is no rule for the log of a sum. Only $\log(a \times b) = \log a + \log b$ . Similarly, $\log(a - b) \neq \log a - \log b$ . The rules only work with products, quotients, and powers — never with sums or differences. This is the single most common log mistake across all levels.

\log_a(mn) = \log_a m + \log_a n

\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n

\log_a(m^n) = n\log_a m

\log_a b = \frac{\log_c b}{\log_c a} = \frac{1}{\log_b a}

\log_a a = 1, \quad \log_a 1 = 0, \quad a^{\log_a x} = x

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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