Problem 1 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, correct to four decimal places. $$ 10^{x}=25 $$

Step-by-Step Solution

Answer

The solution is approximately \(x = 1.3979\).

Step 1: Rewrite the Equation

First, we start with the equation \(10^x = 25\). Our goal is to solve for \(x\). To do this, we will convert the exponential equation into a logarithmic equation.

Step 2: Convert to Logarithmic Form

To solve for \(x\), take the logarithm of both sides of the equation. This gives us: \[ \log_{10}(10^x) = \log_{10}(25) \]

Step 3: Apply Logarithmic Identity

Apply the identity \(\log_{10}(10^x) = x\) because the log base matches the base of the exponential. Thus, the equation becomes simpler:\[ x = \log_{10}(25) \]

Step 4: Calculate the Logarithm

Evaluate the logarithm \(\log_{10}(25)\). You can use a calculator to do this. Input \(25\) into the base-10 logarithm function on your calculator to get \[ x \approx 1.3979 \]Round your answer to four decimal places, ensuring precision.

Key Concepts

Logarithmic Form

Understanding logarithmic form is key when solving exponential equations. Logarithms provide a way to express exponential equations in a different, often simpler, form. In an exponential equation such as \(10^x = 25\), the unknown \(x\) represents the power to which the base of 10 is raised. By converting this to logarithmic form, you utilize the relationship between exponents and logarithms, making it easier to isolate and solve for \(x\).

Here's how it works:

• The base of the exponential (10 in this case) becomes the base of the logarithm.
• The exponent \(x\) becomes the result of the logarithm.
• The result of the exponentiation (25) becomes the argument of the logarithm.

Thus, the equation \(10^x = 25\) is rewritten in logarithmic form as \(\log_{10}(25) = x\). This form is much easier to handle with mathematical tools, especially calculators, to find the precise value of \(x\).

Logarithmic Identity

Logarithmic identities are simple rules that help simplify logarithms. The specific identity used in solving our equation \(10^x = 25\) is based on the inverse relationship between logarithms and exponential functions. The core identity applied here is \(\log_b(b^x) = x\).

This identity states that taking the logarithm of a power with a matching base simply returns the exponent. In our example, since our base is 10, we use \(\log_{10}(10^x) = x\). This dramatically simplifies the process of finding the unknown \(x\), because it allows us to remove the exponent and logarithm simultaneously, effectively peeling back the layers to uncover \(x\).

• The base of the logarithm must match the base of the exponent for the identity to apply.
• Once the identity is applied, solving for \(x\) becomes a matter of calculating a simple logarithm on the other side (\(\log_{10}(25)\)).

Simple yet powerful, such identities are the backbone of handling more complex logarithmic expressions efficiently.

Base-10 Logarithm

The base-10 logarithm, also known as the common logarithm, is one of the most widely used logarithmic forms, especially in higher education and scientific contexts. Represented as \(\log_{10}\) or simply \(\log\) in many calculators and textbooks, this common logarithm is designed to simplify calculations involving powers of 10.

When dealing with the equation \(10^x = 25\), using base-10 logarithms reduces complications because the base matches the exponential base:

• The equation \(\log_{10}(10^x) = x\) shows the utility of this logarithm under our current context.
• Even calculators default to base-10 logarithms, facilitating quick calculations.

The base-10 logarithm also has extensive real-world applications, such as measuring sound intensity (decibels), orders of magnitude (such as pH levels), and in financial calculations.

To find \(x\), simply enter \(25\) into the common logarithm function of a calculator, resulting in approximately \(1.3979\) when rounded to four decimal places. Understanding base-10 logarithms not only helps solve equations elegantly but also bridges abstract math concepts to practical applications.