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Chapter 4: Problem 116

Find the approximate solution to each equation. Round to four decimal places. $$10^{3 x}=5$$

Short Answer

Expert verified
x 鈮� 0.2330

Step by step solution

01

Apply Logarithms

To solve the equation involving an exponent, apply the logarithm to both sides of the equation. Use the natural logarithm (ln) for simplicity: \[ \ln(10^{3x}) = \ln(5) \]
02

Use Logarithm properties

Apply the logarithm power rule, which states \( \ln(a^b) = b \ln(a) \). This allows the exponent to be moved in front of the logarithm: \[ 3x \ln(10) = \ln(5) \]
03

Solve for x

Isolate the variable x by dividing both sides of the equation by \( 3 \ln(10) \): \[ x = \frac{\ln(5)}{3 \ln(10)} \]
04

Compute the natural logarithms

Use a calculator to find the natural logarithms and then compute the value of x: \[ x = \frac{\ln(5)}{3 \ln(10)} \approx \frac{1.6094}{6.9078} \approx 0.2330 \]
05

Round the answer

Round the result to four decimal places: \[ x \approx 0.2330 \]

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Key Concepts

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

Logarithms
Logarithms are vital in solving equations involving exponential functions. The logarithm of a number is the exponent to which the base must be raised to yield that number. For example, if we have base 10, \( \text{log}_{10}(100) = 2 \) because 10 raised to the power of 2 equals 100. In our problem, we apply logarithms to both sides to simplify the exponential form of the equation.
By converting the exponent into a more manageable form, logarithms make it easier to isolate variables. This is why step 1 of the solution involves applying a logarithm to both sides of the equation.
Natural Logarithm
The natural logarithm, denoted as \( \text{ln} \), uses Euler's number (approximately 2.71828) as the base. This type of logarithm is particularly useful in continuous growth and decay problems. In our exercise, we chose the natural logarithm over the common logarithm (base 10), although either would work.
Using \( \text{ln} \) is often simpler in practice because of the ease of computation with many calculators and various mathematical properties. When we take the natural logarithm of both sides, we essentially transform the original complex equation into a linear equation that we can solve using algebraic methods.
Exponent Properties
Understanding exponent properties is crucial when dealing with logarithms. A key property used in the solution is the logarithm power rule: \( \text{ln}(a^b) = b \text{ln}(a) \). This property allows us to bring the exponent in front of the logarithm as a multiplier, simplifying the equation.
Another important property is \( a^{m/n} = (\sqrt[n]{a})^m \). Knowing how to manipulate and combine exponents can significantly ease solving exponential equations. In our problem, using these properties allows us to transform the exponential component into a linear form: \( 3x \text{ln}(10) = \text{ln}(5) \).
Rounding Numbers
Rounding numbers ensures the final answer is presented in a manageable and consistent form. In mathematics, rounding to a specified number of decimal places can make results more comprehensible and tidy. For our solution, we round to four decimal places as instructed.
After calculating \( x \), we get an approximate value from the natural logarithms: \( x \approx 0.2329921 \). To present this appropriately, we round it to four decimal places: \( x \approx 0.2330 \). Always ensure to follow rounding rules: if the digit after the desired decimal place is 5 or greater, round up the last significant digit.

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Most popular questions from this chapter

Solve each problem. When needed, use 365 days per year and 30 days per month. Periodic Compounding A deposit of \(\$ 5000\) earns \(8 \%\) annual interest. Find the amount in the account at the end of 6 years and the amount of interest earned during the 6 years if the interest is compounded a. annually b. quarterly c. monthly d. daily.

Room Temperature Marlene brought a can of polyurethane varnish that was stored at \(40^{\circ} \mathrm{F}\) into her shop, where the temperature was \(74^{\circ} .\) After 2 hr the temperature of the varnish was \(58^{\circ} .\) If the varnish must be \(68^{\circ}\) for best results, then how much longer must Marlene wait until she uses the varnish?

Solve each problem. The population of the world doubled from 1950 to \(1987,\) going from 2.5 billion to 5 billion people. Using the exponential model, $$P=P_{0} e^{r t},$$ find the annual growth rate \(r\) for that period. Although the annual growth rate has declined slightly to \(1.63 \%\) annually, the population of the world is still growing at a tremendous rate. Using the initial population of 5 billion in 1987 and an annual rate of \(1.63 \%\), estimate the world population in the year 2010

Consider the function \(y=\log \left(10^{n} \cdot x\right)\) where \(n\) is an integer. Use a graphing calculator to graph this function for several choices of \(n .\) Make a conjecture about the relationship between the graph of \(y=\log \left(10^{n} \cdot x\right)\) and the graph of \(y=\log (x) .\) Save your conjecture and attempt to prove it after you have studied the properties of logarithms, which are coming in Section \(4.3 .\) Repeat this exercise with \(y=\log \left(x^{n}\right)\) where \(n\) is an integer.

Solve each problem. At what interest rate would a deposit of \(\$ 30,000\) grow to \(\$ 2,540,689\) in 40 years with continuous compounding?
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