Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 55

Chapter 5, Problem 55

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log₂(x+25)=4

Verified step by step guidance

Identify the given logarithmic equation: $\log_{2}(x + 25) = 4$.

Recall the definition of logarithm: $\log_{b}(A) = C$ means $b^{C} = A$. Apply this to rewrite the equation as $2^{4} = x + 25$.

Calculate \$2^{4}$ (without final numeric evaluation here) and set up the equation $x + 25 = 2^{4}$.

Solve for $x$ by isolating it: $x = 2^{4} - 25$.

Check the domain restriction for the logarithm: the argument $x + 25$ must be greater than 0, so ensure $x + 25 > 0$ and verify that your solution satisfies this condition.

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

Understanding the basic properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations in exponential form to solve for the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires that the argument x be positive. When solving equations, it is crucial to check that solutions do not make the argument of any logarithm zero or negative, as these are not valid.

Exact and Approximate Solutions

After finding the exact solution to a logarithmic equation, it is often necessary to provide a decimal approximation. Using a calculator to round the solution to a specified number of decimal places helps interpret and communicate the result clearly.

Recommended video:

4:20

Graph Hyperbolas at the Origin

Related Practice

Textbook Question

In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is $1$ . $log 3 minus 3 log x$

1176

views

Textbook Question

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. 5 ln x - 2 ln y

1059

views

Textbook Question

Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x) = 2 + log₂x

1057

views

Textbook Question

Use the compound interest formulas A = P (1+ r/n)^nt and A =Pe^rt to solve exercises 53-56. Round answers to the nearest cent. Find the accumulated value of an investment of \$10,000 for 5 years at an interest rate of 1.32% if the money is d. compounded continuously.

2908

views

Textbook Question

Use the compound interest formulas A = P (1+ r/n)^nt and A =Pe^rt to solve exercises 53-56. Round answers to the nearest cent. Suppose that you have \$12,000 to invest. Which investment yields the greater return over 3 years: 0.96% compounded monthly or 0.95% compounded continuously?