Skip to main content
Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 29
Chapter 5, Problem 29

Evaluate each expression without using a calculator. log7 √7

Verified step by step guidance
1
Recognize that the expression is \( \log_7 \sqrt{7} \), which means the logarithm base 7 of the square root of 7.
Rewrite the square root of 7 using an exponent: \( \sqrt{7} = 7^{\frac{1}{2}} \).
Substitute this back into the logarithm: \( \log_7 7^{\frac{1}{2}} \).
Use the logarithm power rule, which states \( \log_b (a^c) = c \cdot \log_b a \), to simplify: \( \log_7 7^{\frac{1}{2}} = \frac{1}{2} \cdot \log_7 7 \).
Since \( \log_7 7 = 1 \) (because any log base of itself is 1), the expression simplifies to \( \frac{1}{2} \cdot 1 = \frac{1}{2} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product, quotient, and power rules. For example, the power rule states that log_b(a^c) = c * log_b(a), which helps in rewriting and evaluating logarithmic expressions without a calculator.
Recommended video:
5:36
Change of Base Property

Understanding Radicals as Exponents

A square root can be expressed as an exponent of 1/2, so √7 is equivalent to 7^(1/2). This conversion allows the use of exponent rules within logarithmic expressions, making it easier to simplify and evaluate the expression.
Recommended video:
Guided course
04:06
Rational Exponents

Logarithm of the Base

The logarithm of a base to itself is always 1, meaning log_b(b) = 1. This fundamental fact is essential when simplifying expressions like log_7(7^(1/2)), as it directly leads to the evaluation of the logarithm.
Recommended video:
7:30
Logarithms Introduction
Related Practice
Textbook Question

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. h(x) = 2x+1 – 1

841
views
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 √(100x)

1473
views
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. 5x=17

704
views
Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb ((x2y)/z2)

1186
views
Textbook Question

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2x – 1

884
views
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. 5ex=23

711
views