Problem 76 - Algebra 2

Question

Solve each equation. Round to the nearest hundredth. \(2^{x}=10\)

Step-by-Step Solution

Answer

The value of \(x\) is approximately 3.32, rounded to the nearest hundredth.

Step 1: Understand the Equation

We need to find the value of \(x\) that satisfies the equation \(2^{x} = 10\). Since this equation involves an exponent with a base of 2 leading to a non-power of 2 (i.e., 10), we'll need logarithms to solve it.

Step 2: Apply Logarithms

To solve the equation, take the logarithm of both sides. We can use either natural logarithm (ln) or common logarithm (log). For instance, we'll use the natural logarithm. The equation becomes \(\ln(2^{x}) = \ln(10)\).

Step 3: Use Logarithm Properties

Apply the power rule of logarithms to simplify: \(x \cdot \ln(2) = \ln(10)\). The power rule states that \(\ln(a^b) = b\cdot \ln(a)\).

Step 4: Solve for x

Rearrange the equation to solve for \(x\): \(x = \frac{\ln(10)}{\ln(2)}\).

Step 5: Calculate the Result

Using a calculator, evaluate the right side of the equation. We approximate \(\ln(10) \approx 2.302\) and \(\ln(2) \approx 0.693\). Then, \(x \approx \frac{2.302}{0.693}\).

Step 6: Round the Answer

Calculate the division \(\frac{2.302}{0.693}\) which results in \(x \approx 3.319\). Round this to the nearest hundredth to get \(x \approx 3.32\).

Key Concepts

Exponential Equations

Exponential equations have variables positioned in the exponent like in our example: \(2^{x} = 10\). The core challenge of these equations is finding the value of the variable that makes the equation true.
These equations are prevalent in fields such as finance, physics, and computer science. They often represent growth or decay processes, such as population expansion or radioactive decay. Recognizing the pattern of an exponential equation helps you quickly identify the type of solution approach required.
To solve exponential equations effectively, understanding the relationship between exponents and logarithms is essential. Solvers frequently rely on logarithms when the exponential equation cannot be simplified by common powers. Let's dive deeper into natural logarithms to understand how they help solve such equations.

Natural Logarithms

Natural logarithms are a specific type of logarithm using the base \(e\), where \(e\) is an irrational constant approximately equal to 2.718. They are denoted by \(\ln\) and are frequently used in mathematical and scientific applications.
Natural logarithms simplify the process of solving exponential equations where the base is not easily reduced to simpler expressions. By taking the natural logarithm of both sides, the variable in the exponent can be brought down to a manageable level.
For example, when we have \(\ln(2^{x}) = \ln(10)\), using the power rule of logarithms eases the complexity. It allows us to break down the expression into a simple linear equation, making calculations regarding unknown variables more accessible.
Understanding when and how to use natural logarithms is a significant step toward making challenging equations more manageable.

Logarithm Properties

Logarithm properties are crucial for simplifying equations and solving for unknowns. These properties include the product rule, quotient rule, and power rule, each playing a unique role in equation manipulation.
In solving \(2^{x} = 10\), the power rule is particularly valuable. It states that \(\log(a^b) = b \cdot \log(a)\), thereby transforming an exponentiated variable into a coefficient. This was seen in the step \(x \cdot \ln(2) = \ln(10)\), greatly simplifying further calculations.
Remember, these properties provide powerful shortcuts for handling complex logarithmic expressions. By understanding these principles, like converting multiplication within logarithms to addition, or division to subtraction, you unlock the potential to solve logarithmic equations with ease and efficiency.