/*! 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 46 Solve each logarithmic equation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 46

Solve each logarithmic equation. $$\log _{2}(3 n+7)=5$$

Short Answer

Expert verified
The short answer for the given logarithmic equation is: \(n = \frac{25}{3}\).

Step by step solution

01

Rewriting the equation in exponential form

To solve this logarithmic equation, we should first rewrite the equation in exponential form. Recall that \(\log_{b}(a) = c\) can be rewritten as \(b^{c} = a\). Applying this to our given equation, we have: \(2^{5} = 3n + 7\)
02

Simplifying the equation

Now, we'll simplify the equation by calculating the value of \(2^{5}\): \(32 = 3n + 7\) Next, we want to isolate the variable 'n'. To do that, we should subtract 7 from both sides of the equation: \(32 - 7 = 3n + 7 - 7\) This simplifies to: \(25 = 3n\)
03

Solving for n

To solve for n, we will now divide both sides of the equation by 3: \(\frac{25}{3} = \frac{3n}{3}\) This gives us the value of n: \(n = \frac{25}{3}\) So, the solution to the given logarithmic equation is: $$n = \frac{25}{3}$$

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.

Exponential Form
Converting a logarithmic equation to exponential form is essential for solving it. Let鈥檚 look at the equation: \(\log_{2}(3n+7) = 5\). The basic principle here is using the fact that if \(\log_{b}(a) = c\), it can be rewritten in exponential form as \(b^{c} = a\). This transformation helps bridge the gap between a logarithmic and an algebraic expression.

By using this property, our equation turns into \(2^{5} = 3n + 7\). This transformation helps because we can now work more easily with the equation to isolate and solve for the variable \(n\). Understanding how to rewrite from log to exponential form can demystify logarithmic equations, making them less intimidating.
Isolate the Variable
To solve the equation, we need to isolate the variable \(n\). Isolating the variable means getting \(n\) by itself on one side of the equation. Once your equation is in the form \(2^{5} = 3n + 7\), the next step is to remove constants from \(n\). To do so:

  • Subtract 7 from both sides to ensure that \(3n\) is by itself on one side of the equation.
This gives us: \(32 - 7 = 3n\), simplifying to \(25 = 3n\). Isolating the variable helps focus our efforts on solving for \(n\) and is crucial in breaking down equations.
Solve for n
After isolating \(n\), the next step is to actually solve for it. We have the equation \(25 = 3n\). To solve for \(n\):

  • Divide each side by 3 to remove the coefficient from \(n\).
By doing this, we find that \(n = \frac{25}{3}\). Remember, solving for a variable often involves simple operations like division or multiplication to separate the variable from other numbers.
Properties of Logarithms
Understanding the properties of logarithms is vital, as these properties allow us to manipulate and solve logarithmic equations more effectively. Some key properties include:
  • \(\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\), detailing how logs convert multiplication into addition.
  • \(\log_{b}(\frac{m}{n}) = \log_{b}(m) - \log_{b}(n)\), explaining how division inside a log converts to subtraction of logs.
  • \(\log_{b}(m^{c}) = c \cdot \log_{b}(m)\), illustrating how exponents can 'come down'.
Although this particular problem didn鈥檛 require those properties, having them in your toolkit is extremely useful. It's the understanding of these basics that can help tackle more complex equations.

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

Show that the inverse of \(y=\ln x\) is \(y=e^{x}\).

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\log \left(r^{2}+3\right)-2 \log \left(r^{2}-3\right)$$

Radioactive carbon- 14 is a substance found in all living organisms. After the organism dies, the carbon- 14 decays according to the equation $$y=y_{0} e^{-0.000121 t}$$ where \(t\) is in years, \(y_{0}\) is the initial amount present at time \(t=0,\) and \(y\) is the amount present after \(t\) yr. a) If a sample initially contains 15 g of carbon- 14 how many grams will be present after 2000 yr? b) How long would it take for the initial amount to decay to 10 g? c) What is the half-life of carbon- \(14 ?\)

Determine whether each function is one-to-one. If it is one-to-one, find its inverse. $$g=\\{(2,1),(5,2),(7,14),(10,19)\\}$$

The hydronium ion concentrations, \(\left[\mathrm{H}^{+}\right],\) are given for some common substances. Find the \(\mathrm{pH}\) of each substance (to the tenths place), and determine whether each substance is acidic or basic. $$\text { Tomatoes: }\left[\mathrm{H}^{+}\right]=1 \times 10^{-4}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

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.