Problem 22 - Precalculus

Question

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$3\left(2^{x}\right)+8=35$$

Step-by-Step Solution

Answer

\(x \approx 3.170\)

Step 1: Isolate the exponential term

To start, subtract 8 from both sides of the equation: \(3(2^x) + 8 = 35\) becomes \(3(2^x) = 27\).

Step 2: Solve for the base term

Next, divide both sides by 3 to isolate \(2^x\): \(2^x = 9\).

Step 3: Solve for the exponent using logarithms

To solve for \(x\), take the logarithm of both sides. Using the base 2 logarithm, the equation becomes \(x = \log_2(9)\).

Step 4: Convert using common logarithms

Convert the logarithm to the common logarithm (base 10) using the change of base formula: \(x = \frac{\log_{10}(9)}{\log_{10}(2)}\).

Step 5: Calculate the solution

Use a calculator to find \(x\). It yields approximately \(x = 3.170\).

Key Concepts

Logarithms

Logarithms are an essential tool in mathematics, particularly when solving exponential equations. They help transform multiplicative relationships into additive ones, making complex equations simpler to handle. The basic idea of a logarithm is to ask, "To what power must a certain number (called the base) be raised, to obtain another number?" For example, if you have a base 10 logarithm, written as \( \log_{10}(100) \), it means "What power do you raise 10 to, to get 100?" The answer, in this case, is 2, because \(10^2 = 100\).
Logarithms are widely used due to their ability to solve equations where the unknown appears as an exponent. In such cases, taking the logarithm of both sides of an equation can simplify the process. Understanding the properties of logarithms, such as the product, quotient, and power rules, further aids in their usage, providing versatility in various mathematical applications.

Isolate Exponential Term

Isolating the exponential term is a critical first step in solving exponential equations. This process involves manipulating the equation so that the term with the exponent stands alone on one side. Take the original problem: \(3(2^x) + 8 = 35\). Our goal is to simplify this equation to have the form \(b^x = A\) where \(b\) is the base, and \(A\) is a constant.
For our exercise, subtract 8 from both sides to remove the extra constant, transforming the equation to \(3(2^x) = 27\). Next, divide both sides by 3 to ensure that the exponential term is completely isolated: \(2^x = 9\). Isolating this term simplifies solving for \(x\) because it focuses attention solely on the variables involved in the exponential expression.

• First, remove constants outside the exponential term by using addition or subtraction.
• Second, eliminate any coefficients next to the exponential term using division or multiplication.
• Finally, ensure the exponential term is in the form \(b^x = A\).

This method organizes the equation, allowing for a straightforward application of logarithms in the next steps.

Change of Base Formula

The change of base formula is a useful technique for evaluating logarithms when the base is not convenient or accessible on a standard calculator. In the exercise, we use this formula to convert a base 2 logarithm into a base 10 logarithm, which is supported by most calculators.
The change of base formula is:\[\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\]where \(b\) is the original base, \(c\) is the new base, and \(a\) is the number for which you're taking the logarithm. In our example, \(x = \log_2(9)\) is converted using base 10 as follows: \(x = \frac{\log_{10}(9)}{\log_{10}(2)}\).
By using the change of base formula, you can solve for \(x\) using basic logarithms, making exponential equations more manageable. This approach is extremely practical as it allows us to work with any logarithmic base preferred, ensuring flexibility in solving a broader range of mathematical problems.