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

Recall the logarithmic property that states the sum of two logarithms with the same base can be written as the logarithm of the product: $\log a + \log b = \log (a \times b)$.

Identify the terms in the expression: $\log 5 + \log 2$ both have the same base (common logarithm, base 10).

Apply the property to combine the two logarithms into one: $\log (5 \times 2)$.

Simplify the product inside the logarithm: $\log 10$.

Recognize that $\log 10$ (base 10) equals 1, so the expression simplifies to $1$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Recommended video:

5:14

Evaluate Logarithms

Related Practice

Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log$ $\left$( $\frac{10x^2 \sqrt[3]{1 - x}$}{7(x + 1)^2} $\right$)$

Textbook Question

Evaluate each expression without using a calculator. $8^{$\log$_8 19}$

Textbook Question

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = ln x and g(x) = - ln (2x)

Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 7^0.3x=813

Textbook Question

The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e^-x

Textbook Question