/*! 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 75 Use a calculator to solve each e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 75

Use a calculator to solve each equation. Round answers to four decimal places. $$3=10^{x}$$

Short Answer

Expert verified
x 鈮� 0.4771

Step by step solution

01

Rewrite the Equation

Rewrite the given equation in logarithmic form so we can isolate the variable x. Given the equation is: \[ 3 = 10^x \] Rewrite it as: \[ x = \text{log}_{10}(3) \]
02

Use a Calculator

Use a scientific calculator to calculate the logarithm base 10 of 3. On most scientific calculators, you can do this by pressing the 'log' button and then entering '3'. This gives us: \[ x \approx \text{log}_{10}(3) \approx 0.4771 \]
03

Round the Answer

Round the answer to four decimal places. Since the calculator might give more digits, round it to: \[ x \approx 0.4771 \]

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 Logarithms
Logarithms are a fundamental part of mathematics that help us work with exponential equations. To understand logarithms, you need to grasp both their definition and their function. A logarithm essentially tells us the power to which a number must be raised to obtain another number. For example, if we have \(10^x = 3\), we want to know what power (x) we need to raise 10 to get 3. The logarithmic form of this equation is \(x = \text{log}_{10}(3)\). The base of the logarithm (in this case, 10) is the same base as the exponent base in the original equation. Think of logarithms as the reverse operation of exponentiation. Here, we are solving for the exponent x by using the logarithm.
Exponents and Their Role
An exponent refers to the number of times a base number is multiplied by itself. For example, in \(10^x\), 10 is the base and x is the exponent. If x is 2, then \(10^2 = 100\). When dealing with logarithmic equations like \(3 = 10^x\), solving for x means finding out the exponent. Exponents and logarithms are inversely related. This means that calculating a logarithm will help you find the exponent when you know the base and the result. Always remember: \( \text{log}_b(a) = c \) is equivalent to \( b^c = a \). In the exercise, we converted the equation to logarithmic form for easier calculation.
Using a Scientific Calculator
A scientific calculator can make solving logarithmic equations much easier. These calculators come with a logarithm function, usually labeled as 'log'. To solve \(x = \text{log}_{10}(3)\) on a scientific calculator:
  • First, press the 'log' button on your calculator.
  • Then, enter the number 3.
  • The calculator will display the result, which is approximately 0.4771 in this case.
Make sure your calculator is in the correct mode (most often, common log mode) to handle base 10 logarithms correctly. Scientific calculators are powerful tools for many math problems, so getting familiar with their functions is very useful for students.

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

In chemistry the \(p H\) of a solution is defined by $$\mathrm{pH}=-\log _{10}[H+]$$ where \(H+\) is the hydrogen ion concentration of the solution in moles per liter. Distilled water has a pH of approximately \(7 . A\) solution with a pH under 7 is called an acid, and one with a pH over 7 is called a base. Roanoke River pH. The hydrogen ion concentration of a water sample from the Roanoke River at Janesville, North Carolina, was \(1.995 \times 10^{-7} \mathrm{mol} / \mathrm{L}\) (wwwnc.usgs.gov). What was the pH of this water sample?

Depreciating knowledge. The value of a certain textbook seems to decrease according to the formula \(V=45 \cdot 2^{-0.9 t}\) where \(V\) is the value in dollars and \(l\) is the age of the book in years. What is the book worth when it is new? What is it worth when it is 2 years old?

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 $$3^{x}=20$$

Solve each problem. The number of grams of a radioactive substance that is present in an old piece of cloth after t years is given by $$A=10 e^{-0.0001 t}$$ How many grams of the radioactive substance did the cloth contain when it was made at time \(t=0 ?\) If the cloth now contains only 4 grams of the substance then when was the cloth made?

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 $$x \cdot \log (3)=\log (5)$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Pure Maths

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.