Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Introduction to Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Introduction to Logarithms

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1:05 minutes

Problem 99

Textbook Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. 10^(log √x)

Verified step by step guidance

Recognize that the expression is in the form of \( a^{\log_b(c)} \).

Recall the property of logarithms: \( a^{\log_a(b)} = b \).

In this case, \( a = 10 \), \( b = \sqrt{x} \), and the base of the logarithm is also 10.

Apply the property: \( 10^{\log_{10}(\sqrt{x})} = \sqrt{x} \).

Conclude that the expression simplifies to \( \sqrt{x} \).

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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. One key property is that log(a^b) = b * log(a). This means that logarithmic expressions can often be rewritten to make calculations easier, especially when dealing with exponents and roots.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which can be useful in simplifying expressions. For example, log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This is particularly helpful when evaluating logarithms with bases that are not easily computable.

Exponential and Logarithmic Relationships

Exponential functions and logarithmic functions are inverses of each other. This means that if y = b^x, then x = log_b(y). Understanding this relationship is crucial for simplifying expressions involving both exponentials and logarithms, as it allows us to switch between the two forms effectively.