Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take \(\ln\) of both sides, obtaining \(x \ln 3=\ln 11 ;\) then \(x=(\ln 11) /(\ln 3) \approx 2.1827 .\) $$ 5^{x}=13 $$

Step-by-Step Solution

Answer

The solution is approximately \(x = 1.5790\).

Step 1: Recognize the Equation

We have the equation \(5^x = 13\) which is an exponential equation where the base is \(5\) and the exponential term is \(x\). We need to solve for \(x\).

Step 2: Apply the Natural Logarithm

Take the natural logarithm of both sides of the equation to bring down the exponent. We apply \(\ln\) to both sides of \(5^x = 13\) resulting in \(\ln(5^x) = \ln(13)\).

Step 3: Utilize Logarithmic Identity

Use the property of logarithms that allows the exponent to come out: \(\ln(a^b) = b \cdot \ln(a)\). Apply this to get \(x \cdot \ln(5) = \ln(13)\).

Step 4: Isolate x

Divide both sides of the equation by \(\ln(5)\) to solve for \(x\). Hence, \(x = \frac{\ln(13)}{\ln(5)}\).

Step 5: Calculate the Value

Substitute the logarithm values into the equation using a calculator: \(x \approx \frac{2.5649}{1.6094} \approx 1.5790\).

Key Concepts

Exponential Equations

Exponential equations are equations where variables appear as exponents. These types of equations are solved using different techniques compared to linear or quadratic equations, due to the positions of the variables. Let's explore an example:

• For a basic exponential equation like \(5^x = 13\), the variable \(x\) is in the exponent position.


• The equation involves a constant base (5 in this case), raised to the power of a variable \(x\), and is set equal to another constant (13 here).


• Traditional algebraic methods won't directly help since the variable is an exponent, hence the need for logarithms, particularly natural logarithms.

This form is typical because it stems from the properties of exponential functions, which rapidly increase or decrease. These equations are very common in fields like science and finance, modeling exponential growth and decay.

Properties of Logarithms

To solve exponential equations like \(5^x = 13\), we use logarithms because of their ability to "bring down the exponent." This is one of several crucial properties of logarithms.Logarithms, specifically natural logarithms (denoted as \(\ln\)), transform multiplicative relationships into additive ones, making them very useful. The properties you need to know include:

• Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\)


• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)


• Power Rule: \(\ln(a^b) = b \cdot \ln(a)\)

The power rule, particularly, is instrumental in solving exponential equations because it allows us to switch the exponent to a multiplier, which simplifies our task when isolating the variable.

Solving Logarithmic Equations

Once the properties of logarithms are understood, we can apply them to solve equations like \(5^x = 13\). Here's the step-by-step method:First, use the natural logarithm on both sides of the equation:

• Start with \(\ln(5^x) = \ln(13)\).


• Applying the power rule (\(\ln(a^b) = b \cdot \ln(a)\)), bring down \(x\) from the exponent: \(x \cdot \ln(5) = \ln(13)\).


• To solve for \(x\), divide both sides by \(\ln(5)\): \(x = \frac{\ln(13)}{\ln(5)}\).


• Use a calculator to get a numerical approximation. For this example, \(\ln(13) \approx 2.5649\) and \(\ln(5) \approx 1.6094\), giving us \(x \approx 1.5790\).

By applying the properties of logarithms strategically, we turn a complex exponential equation into a simple arithmetic problem, making it solvable by basic division.