Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log 5 + log 2

Verified step by step guidance

1. Recall the logarithmic property that states the sum of two logarithms with the same base can be written as the logarithm of the product: \(\log a + \log b = \log (a \times b)\).
2. Identify the terms in the expression: \(\log 5 + \log 2\) both have the same base (common logarithm, base 10).
3. Apply the property to combine the two logarithms into one: \(\log (5 \times 2)\).
4. Simplify the product inside the logarithm: \(\log 10\).
5. Recognize that \(\log 10\) (base 10) equals 1, so the expression simplifies to \$1$.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. The product rule states that log(a) + log(b) = log(ab), allowing the combination of sums into a single logarithm. Understanding these properties is essential for condensing expressions into one logarithm.

Logarithmic Expression Simplification

Simplifying logarithmic expressions involves applying logarithm properties to rewrite sums or differences as a single logarithm. This process often includes removing coefficients by converting them into exponents and combining terms to achieve a single log with coefficient 1.

Evaluating Logarithms Without a Calculator

Some logarithmic expressions can be evaluated exactly by recognizing values and their relationships, such as log base 10 of 5 and 2. Multiplying inside the log can yield a number whose log is known or easily simplified, enabling evaluation without a calculator.