/*! 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 15 Determine the \(x\) -intercept o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 15

Determine the \(x\) -intercept of \(y=\log _{7}(x+2)\).

Short Answer

Expert verified
The x-intercept is at \( x = -1 \).

Step by step solution

01

Understand the x-intercept

The x-intercept is the point where the graph of the equation crosses the x-axis. At the x-intercept, the value of y is 0.
02

Set y to 0 in the equation

Given the equation:\[ y = \log _{7}(x+2) \]. To find the x-intercept, set y to 0:\[ 0 = \log _{7}(x+2) \].
03

Solve the logarithmic equation

To solve \log _{7}(x+2) = 0, rewrite the equation in exponential form: \( 7^0 = x+2 \).
04

Simplify the exponential equation

Since any number raised to the power of 0 is 1, we get: \( 1 = x + 2 \).
05

Solve for x

Subtract 2 from both sides of the equation to find x: \( x = 1 - 2 \), which simplifies to \( x = -1 \).
06

Verify the solution

Substitute \( x = -1\) back into the original equation to ensure it satisfies \( y = \log _{7}(x+2) \): \( y = \log _{7}((-1)+2) = \log _{7}(1) = 0 \), confirming that the solution is correct.

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.

Understanding x-intercepts
To understand the x-intercept of a function, you need to remember that it is the point where the graph touches or crosses the x-axis. At this point, the value of the function, which is y, is always zero.
In simpler terms, for any equation, if we set y to zero, we can find where the equation intersects the x-axis. For example, in the equation given in the problem, we set y to 0 to find the x-intercept of the function.
Solving logarithmic equations
Logarithmic equations may seem complicated at first, but they're manageable if you follow each step carefully. First, understand the properties of logarithms. A logarithm answers the question: 'To what exponent must we raise a certain base to obtain a given number?'.
Let's take the exercise's equation: \( y = \log _{7}(x+2) \). To find the x-intercept, we set y to 0. This gives us \( 0 = \log _{7}(x+2) \). This means the exponent that makes the base 7 equal to \( x+2 \) is zero. To solve this, we rewrite it in exponential form.
Rewriting in exponential form
Rewriting logarithmic equations in exponential form can simplify solving them. The rule here is: if \( \log _{b}(a) = c \), this is equivalent to saying \( b^c = a \).
In our case, \( \log _{7}(x+2) = 0 \), that's equivalent to \( 7^0 = x+2 \). We know any number to the power of 0 is 1, so we get \( 1 = x+2 \). Now, it simplifies to solving the simple linear equation: Subtract 2 from both sides to get \( x = -1 \). This process simplifies solving logarithmic functions and makes equations easier to handle.

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

The formula for the Richter magnitude, \(M\) of an earthquake is \(M=\log \frac{A}{A_{0}},\) where \(A\) is the amplitude of the ground motion and \(A_{0}\) is the amplitude of a standard earthquake. In \(1985,\) an earthquake with magnitude 6.9 on the Richter scale was recorded in the Nahanni region of the Northwest Territories. The largest recorded earthquake in Saskatchewan occurred in 1982 near the town of Big Beaver. It had a magnitude of 3.9 on the Richter scale. How many times as great as the seismic shaking of the Saskatchewan earthquake was that of the Nahanni earthquake?

Express in exponential form. a) \(\log _{5} 25=2\) b) \(\log _{a} 4=\frac{2}{3}\) c) \(\log 1000000=6\) d) \(\log _{11}(x+3)=y\)

Express in logarithmic form. a) \(12^{2}=144\) b) \(8^{\frac{1}{3}}=2\) 101 c) \(10^{-5}=0.000\) d) \(7^{2 x}=y+3\)

Radioisotopes are used to diagnose various illnesses. Iodine-131 (I-131) is administered to a patient to diagnose thyroid gland activity. The original dosage contains 280 MBq of I-131. If none is lost from the body, then after \(6 \mathrm{h}\) there are \(274 \mathrm{MBq}\) of I-131 in the patient's thyroid. What is the half-life of I-131, to the nearest day?

Write each expression in terms of individual logarithms of \(x, y,\) and \(z\). a) \(\log _{7} x y^{2} \sqrt{z}\) b) \(\log _{5}(x y z)^{8}\) c) \(\log \frac{x^{2}}{y \sqrt[3]{z}}\) d) \(\log _{3} x \sqrt{\frac{y}{z}}\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.