/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Solve each equation. $$2^{x-1}=... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 26

Solve each equation. $$2^{x-1}=5$$

Short Answer

Expert verified
\(x = \frac{\text{ln}(5)}{\text{ln}(2)} + 1 \)

Step by step solution

01

Isolate the exponential expression

The given equation is \(2^{x-1}=5\). The exponential expression \(2^{x-1}\) is already isolated.
02

Apply the natural logarithm to both sides

Apply the natural logarithm (ln) to both sides of the equation to help solve for \(x\): \(\text{ln}(2^{x-1}) = \text{ln}(5)\).
03

Use the power rule of logarithms

Use the power rule of logarithms \(\text{ln}(a^b) = b \text{ln}(a)\) to simplify the left side: \((x - 1) \text{ln}(2) = \text{ln}(5)\).
04

Solve for \(x\)

Isolate \(x\) by first dividing both sides by \(\text{ln}(2)\): \((x - 1) = \frac{\text{ln}(5)}{\text{ln}(2)}\). Then add 1 to both sides to solve for \(x\): \(x = \frac{\text{ln}(5)}{\text{ln}(2)} + 1\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithms
When solving exponential equations, natural logarithms (\text{ln}) are really handy. They help us bring the exponent down to a form that's easier to solve. The natural logarithm has a base of the mathematical constant, e, which is approximately 2.718.
To apply the natural logarithm to both sides of an equation like this one: $$2^{x-1}=5$$
We would write: \text{ln}(2^{x-1}) = \text{ln}(5).
This step makes further maneuvering more manageable.
Power Rule of Logarithms
After applying the natural logarithm, the next step is to use the power rule of logarithms. This rule states that \text{ln}(a^b) = b \text{ln}(a).
So, for our problem: $$\text{ln}(2^{x-1})$$ becomes $$(x-1) \text{ln}(2).$$
It simplifies the left side of the equation significantly. This makes it easier to isolate and solve for the variable, which in our case is $$x.$$
Isolating Variables
The final step in solving the equation is isolating the variable $$x.$$ After applying the natural logarithm and using the power rule, you get: \text{(x - 1) \text{ln}(2) = \text{ln}(5)}.
To isolate $$x$$, divide both sides by \text{ln}(2): $$(x - 1) = \frac{\text{ln}(5)}{\text{ln}(2)}.
Finally, add 1 to both sides: $$x = \frac{\text{ln}(5)}{\text{ln}(2)} + 1.$$
This gives the value of $$x$$ that satisfies the original equation. By following these steps, we successfully solve for the unknown exponent.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each equation. $$4^{3 x}=\left(\frac{1}{2}\right)^{1-x}$$$

Use a calculator to evaluate each logarithm. Round answers to four decimal places. $$\ln (0.23)$$

Determine whether each equation is true or false. $$ \log _{2}\left(\frac{5}{2}\right)=\log _{2}(5)-1 $$

For each equation, find the exact solution and an approximate solution when appropriate. Round approximate answers to three decimal places. See the Strategy for Solving Exponential and Logarithmic $$\log _{3}(1-x)+\log _{3}(2 x+13)=3$$

Solve each problem. In the diversity index \(d\) for a certain water sample as $$d=\log _{2}\left(\frac{3 \sqrt[3]{2}}{2}\right)$$ Use the base-change formula and a calculator to calculate the value of \(d .\) Round the answer to four decimal places.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.