Problem 37

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 $x=(\ln 11) /(\ln 3) \approx 2.1827 .$ $$ 2^{x}=17 $$

Step-by-Step Solution

Verified

Answer

The solution is approximately $x \approx 4.0875$.

1Step 1: Apply Natural Logarithm

Take the natural logarithm (ln) of both sides of the equation. This helps to bring the variable exponent down to a solvable level:\[\ln(2^{x}) = \ln(17)\]

2Step 2: Use Logarithm Property

Apply the power rule of logarithms, which allows exponents to be moved in front of the logarithm:\[x \cdot \ln(2) = \ln(17)\]

3Step 3: Solve for x

Isolate $x$ by dividing both sides of the equation by $\ln(2)$:\[x = \frac{\ln(17)}{\ln(2)}\]

4Step 4: Calculate the Value

Use a calculator to compute $\ln(17)$ and $\ln(2)$, then perform the division:\[x \approx \frac{2.8332}{0.6931} \approx 4.0875\]

Key Concepts

Exponential EquationsPower Rule of LogarithmsSolving for a Variable ExponentCalculating Logarithms

Exponential Equations

Exponential equations are types of equations where the variables appear as exponents. A typical example is when you have a base raised to the power of a variable like in the exercise: $2^{x}=17$. Solving these equations often requires converting the exponential form into a logarithmic form because calculating exponential equations directly isn't always straightforward.

These equations model situations with exponential growth or decay, such as population growth and radioactive decay.
Solving them usually involves taking the logarithm of both sides to linearize the equation.

To delve deeper, let's talk about why logarithms are used. Since exponential growth involves multiplying by the same number repeatedly (like $2^x$), it becomes advantageous to use logarithms, as they transform multiplication into addition. By switching from exponential form to logarithmic form, solving for the variable becomes more manageable.

Power Rule of Logarithms

The power rule of logarithms is a crucial rule that allows us to solve exponential equations effectively. This rule states that for any logarithm $ ext{log}_b (a^c)$, it can be simplified to $c \cdot \text{log}_b (a)$. In terms of natural logarithms, it's written as $ ext{ln}(a^c) = c \cdot \text{ln}(a)$. This rule is pivotal because it enables us to manipulate exponents in equations like $\ln(2^{x}) = \ln(17)$. Using the power rule, we can simplify this to $x \cdot \ln(2) = \ln(17)$, which brings the exponent down in front as a coefficient. This transformation is essential for bringing down the exponent to the same level as other terms in the equation, making it easier to solve for the variable.

Solving for a Variable Exponent

To solve for a variable in the exponent, the power rule of logarithms is typically applied first to simplify the equation. Once we've transformed the exponential equation using logarithms, the next step is to isolate the variable. In the exercise $2^x = 17$, applying $\ln$ to both sides gives us $x \cdot \ln(2) = \ln(17)$. The goal is then to isolate $x$. This is done by dividing both sides of the equation by $\ln(2)$, resulting in $x = \frac{\ln(17)}{\ln(2)}$.

This process involves basic algebraic manipulation such as division.
The isolated form $x = \frac{\ln(17)}{\ln(2)}$ makes $x$ ready for computation.

Remember, isolating the variable exponent is all about simplifying and rearranging the equation strategically to solve the problem.

Calculating Logarithms

Calculating logarithms, especially natural logarithms, is a key final step in solving exponential equations. To calculate a natural logarithm, you can use a calculator, as they commonly have a $\ln$ button for this purpose. For the exercise $x = \frac{\ln(17)}{\ln(2)}$, you will find:

$\ln(17)$ is approximately 2.8332.
$\ln(2)$ is approximately 0.6931.

With these calculations:1. Divide $2.8332$ by $0.6931$2. You will find $x \approx 4.0875$This step involves straightforward arithmetic but is crucial as it provides the solution to the problem. Always double-check your calculations to ensure accuracy, especially when dealing with decimal values.

Problem 37

Other exercises in this chapter

Problem 36

Find the limits. $$ \lim _{x \rightarrow 2^{+}} \frac{x^{2}+2 x-8}{x^{2}-4} $$

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Problem 37

Sketch the graph of a function that has domain $[0,2]$ and is continuous on $[0,2)$ but not on $[0,2]$.

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Problem 37

$$ \text { Find } \lim _{x \rightarrow 1}\left(x^{2}-1\right) /|x-1| \text { or state that it does not exist. } $$

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Problem 37

Prove that $\lim _{x \rightarrow c} f(x)=L \Leftrightarrow \lim _{x \rightarrow c}[f(x)-L]=0$.

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