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Chapter 9: Problem 37

Solve. Where appropriate, include approximations to three decimal places. If no solution exists, state this. $$ \log x=1.2 $$

Short Answer

Expert verified
x \approx 15.849.

Step by step solution

01

- Understand the logarithm equation

The given equation is \( \log x = 1.2 \). This implies that we are looking for a value of \( x \) such that the logarithm of \( x \) with base 10 equals 1.2.
02

- Rewrite the logarithm equation in exponential form

Rewrite the logarithm equation \( \log x = 1.2 \) in its exponential form. Recall that \log_b a = c\ is equivalent to \ a = b^c.\ Here, the base \( b \) is 10: \[ x = 10^{1.2} \]
03

- Calculate the value of x

Use a calculator to find \( 10^{1.2} \). The calculation gives: \ 10^{1.2} \approx 15.849 \.

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Key Concepts

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

logarithmic functions
Logarithmic functions are mathematical functions used to solve for the exponent that gives a particular number when considering a specific base. In other words, a logarithm answers the question: 'To what power must the base be raised to produce a given number?' For example, in the equation \(\log_b(a) = c\), \(b\) is the base, \(a\) is the number we want to find the exponent for, and \(c\) is the exponent. So, \(\log_{10}(100) = 2\) means that 10 raised to the power of 2 is 100. Logarithmic functions are particularly useful in situations involving multiplication and exponentiation.
exponential form of logarithms
Understanding the exponential form of logarithms is essential in solving logarithmic equations. To convert a logarithmic equation like \(\log_b(a) = c\) into its exponential form, apply the rule that \(a = b^c\). This conversion simplifies the problem by turning it into an exponentiation problem. Taking the original equation of our problem, \(\log(x) = 1.2\), we can use this rule. Since the base in common logarithms is 10, the equation becomes \(x = 10^{1.2}\). This transformation is critical because it allows us to use our knowledge of exponents and calculators to find solutions more easily.
base 10 logarithms
Base 10 logarithms, also known as common logarithms, use 10 as the base. They are commonly written as \(\log(x)\) instead of \(\log_{10}(x)\). This implies that we are solving for an exponent where 10 raised to that exponent equals the given number. For instance, in our problem \(\log(x) = 1.2\), we are asking: '10 raised to what power equals x?' Base 10 logarithms are widely used in scientific calculations and engineering because they simplify the understanding of large-scale data. They allow for easier manipulation and understanding of exponential growth and decay.
calculator usage for logarithms
Using a calculator to find logarithms or solve equations involving logarithms can simplify the process significantly. Most scientific calculators have a \(\log\) button, which generally refers to base 10. To solve our problem where \(x = 10^{1.2}\), you simply:
  • Turn on your calculator.
  • Press the \(10^{x}\) button, or enter 10, follow it by the exponentiation button \(^y\), and then type 1.2.
  • Press the 'equals' \(=\) button to get the result.
The answer will be approximately 15.849. Calculators handle the complex arithmetic behind the scenes, giving you a quick and accurate result. This is especially helpful for non-integer and irrational results.

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Most popular questions from this chapter

The number of computers infected by a virus \(t\) days after it first appears usually increases exponentially. In 2009 the "Conflicker" worm spread from about 2.4 million computers on January 12 to about 3.2 million computers on January \(13 .\) Data: PC World a) Find the exponential growth rate \(k\) and write an equation for an exponential function that can be used to predict the number of computers infected \(t\) days after January \(12,2009\) b) Assuming exponential growth, estimate how long it took the Conflicker worm to infect 10 million computers.

Determine whether or not the given pairs of functions are inverses of each other. \(f(x)=0.75 x^{2}+2 ; g(x)=\sqrt{\frac{4(x-2)}{3}}\)

The concentration of acetaminophen in the body decreases exponentially after a dosage is given. In one clinical study, adult subjects averaged 11 micrograms \(/\) milliliter \((\mathrm{mcg} / \mathrm{mL})\) of the drug in their blood plasma 1 hr after a 1000 -mg dosage and 2 micrograms / milliliter 6 hr after dosage. Data: tylenolprofessional.com; Mark Knopp, M.D. a) Find the value \(k,\) and write an equation for an exponential function that can be used to predict the concentration of acetaminophen, in micrograms / milliliter, t hours after a 1000 -mg dosage. b) Estimate the concentration of acetaminophen 3 hr after a 1000 -mg dosage. c) To relieve a fever, the concentration of acetaminophen should go no lower than \(4 \mathrm{mcg} / \mathrm{mL}\). After how many hours will a 1000 -mg dosage drop to that level? d) Find the half-life of acetaminophen.

Determine whether or not the given pairs of functions are inverses of each other. \(f(x)=0.8 x^{1 / 2}+5.23\) \(g(x)=1.25\left(x^{2}-5.23\right), x \geq 0\)

Solve Is it easier to find \(x\) given \(x=\log _{9} \frac{1}{3}\) or given \(9^{x}=\frac{1}{3} ?\) Explain your reasoning.
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