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Chapter 4: Problem 30

Use the definition of the logarithmic function to find \(x .\) (a) \(\log _{5} x=4\) (b) \(\log _{10} 0.1=x\)

Short Answer

Expert verified
(a) \(x = 625\); (b) \(x = -1\).

Step by step solution

01

Understanding the Logarithmic Equation

The given equation is \(\log_{5} x = 4\). This tells us that \(x\) is a number such that when 5 is raised to what power gives \(x\), the result is 4.
02

Converting Logarithm to Exponential Form

By the definition of a logarithm, \(\log_{b}(a) = c\) can be rewritten in exponential form as \(b^{c} = a\). Thus, we rewrite \(\log_{5} x = 4\) to \(5^4 = x\).
03

Calculating the Exponential Expression

Calculate \(5^4\) to find \(x\).\5^4 = 5 \times 5 \times 5 \times 5 = 625.
04

Understanding the Logarithmic Equation

The given equation is \(\log_{10} 0.1 = x\). Here, 0.1 is the number we get when 10 is raised to the power of \(x\).
05

Converting Logarithm to Exponential Form

Rewriting the logarithmic statement \(\log_{10} 0.1 = x\) into exponential form gives \(10^x = 0.1\).
06

Solving the Exponential Equation

To solve for \(x\), recognize that 0.1 can be rewritten as \(10^{-1}\) because 10 to the power of -1 is 0.1.\Hence, \(10^x = 10^{-1}\) which implies \(x = -1\).

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Key Concepts

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

Exponential Form
Understanding exponential form is crucial for grasping logarithmic functions. An exponential form expresses numbers where a base is raised to a certain power. It is written as \(b^c = a\), where \(b\) is the base, \(c\) is the exponent, and \(a\) is the result. This form is straightforward once you remember its components:
  • Base \((b)\): The number that is being multiplied.
  • Exponent \((c)\): Tells you how many times to multiply the base by itself.
  • Result \((a)\): The answer you get from the multiplication.
For example, if you have \(5^4 = 625\), 5 is the base, 4 is the exponent, and 625 is the result. Using the exponential form lets you transform complex relationships from logarithmic equations into simpler factors that are easy to calculate.
Logarithmic Equation
A logarithmic equation involves the logarithm of a variable or number to one side, linked to a constant. It is essential because it allows you to determine what exponent will produce a certain number. The general form of a logarithmic equation is \(\log_b(a) = c\), translating to: the power you must raise the base \(b\) to get \(a\) results in \(c\).
  • \(\log_5 x = 4\) becomes \(5^4 = x\), showcasing the conversion from logarithmic to exponential form.
  • \(\log_{10} 0.1 = x\) transforms to \(10^x = 0.1\), helping us grasp how logarithms present problems involving powers of ten.
Logarithmic equations are often used in scientific fields to simplify multiplication, division, and calculations involving large numbers.
Solving Equations
Solving equations involving logarithms often requires conversion to a form that allows computation, usually the exponential form. Here鈥檚 a simple process to follow:
  • Identify the logarithmic equation from the problem statement, like \(\log_{b}(a) = c\).
  • Convert the equation to its exponential form using \(b^c = a\).
  • Solve for the unknown variable using basic arithmetic or exponential rules.
Let's solve \(\log_5 x = 4\) by converting it to \(5^4 = x\). Calculate \(5^4 = 625\), so \(x = 625\). In \(\log_{10} 0.1 = x\), rewriting as \(10^x = 0.1\) suggests matching \(10^x\) with \(10^{-1}\) gives \(x = -1\). This straightforward method enables tackling logarithmic problems efficiently.

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Most popular questions from this chapter

Draw the graph of the function in a suitable viewing rectangle, and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$y=\ln \left(x^{2}-x\right)$$

Suppose that the graph of \(y=2^{x}\) is drawn on a coordinate plane where the unit of measurement is an inch. (a) Show that at a distance 2 ft to the right of the origin the height of the graph is about 265 mi. (b) If the graph of \(y=\log _{2} x\) is drawn on the same set of axes, how far to the right of the origin do we have to go before the height of the curve reaches 2 ft?

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If \(D_{0}\) is the original amount of carbon-14 and \(D\) is the amount remaining, then the artifact's age \(A\) (in years) is given by $$A=-8267 \ln \left(\frac{D}{D_{0}}\right)$$ Find the age of an object if the amount \(D\) of carbon- 14 that remains in the object is \(73 \%\) of the original amount \(D_{0}\).

Growth of an Exponential Function Suppose you are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you? (a) One million dollars at the end of the month (b) Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general, \(2^{n}\) cents on the \(n\) th day

Charging a Battery The rate at which a battery charges is slower the closer the battery is to its maximum charge \(C_{0}\). The time (in hours) required to charge a fully discharged battery to a charge \(C\) is given by $$t=-k \ln \left(1-\frac{C}{C_{0}}\right)$$ where \(k\) is a positive constant that depends on the battery. For a certain battery, \(k=0.25 .\) If this battery is fully discharged, how long will it take to charge to \(90 \%\) of its maximum charge \(C_{0} ?\)
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