Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$10^{x}=8.07$$

Step-by-Step Solution

Answer

By calculating the log expression, the solution to the equation is approximately \(x \approx 0.91\)

Step 1: Express the Equation in Logarithmic Form

Rewrite the given exponential equation in logarithmic form using the conversion \[a^b = c\] to \[\log_a c = b\]. So, the given equation \(10^{x}=8.07\) can be rewritten as \(\log_{10} 8.07 = x\)

Step 2: Find a Decimal Approximation

In order to convert the logarithmic solution obtained in the previous step into a decimal, use a scientific calculator and evaluate the log expression. Remember that \(\log_{10} a\) is commonly known as \(\lg a\), and it can be calculated easily. Thus, \(\lg 8.07\) would give the desired decimal approximation of x.

Step 3: Round off the Decimal

Now, to obtain the solution correct to two decimal places, round off the obtained decimal approximation. The values after the decimal point, if there are more than two, need to be rounded off to the nearest hundredths

Key Concepts

Logarithms

Logarithms serve as the inverse function to exponentiation. When you take the logarithm of a number, you are trying to find the power to which a specific base must be raised to get that number. For instance, the logarithmic form of the equation \(10^x = 8.07\) is expressed as \(\log_{10} 8.07 = x\). This means that 10 raised to the power \(x\) results in 8.07. Most commonly, we use base 10 for logarithms in daily calculations, which is known as the common logarithm and is often denoted as \(\log\) or \(\lg\).

• Inverse Relationship: If you know how much the base must be raised to get a number, you know the logarithm.
• Base Matters: While base 10 is common, logarithms can technically use any positive number as a base, such as base 2 or base \(e\).

Logarithms are powerful tools in simplifying multiplication and division into addition and subtraction.

Decimal Approximation

Once logarithms are calculated, they often need to be expressed as decimal approximations. Unlike integers, decimals can represent more precise values. For example, the logarithmic solution \(\log_{10} 8.07\) needs to be converted into a decimal to solve our equation effectively. Utilizing a scientific calculator makes this task straightforward. Decimals provide a convenient way to work with values that are not whole numbers.

• Convenience: Decimals allow for precise calculations and representations of numbers between integers.
• Purpose: Approximating to a decimal helps in understanding and utilizing the logarithmic result practically.

Thus, by turning logarithmic results into decimals, we make them easy to interpret and apply in further calculations.

Scientific Calculator

A scientific calculator is a vital tool for calculating logarithms, like \(\log_{10} 8.07\). These calculators have special functions to handle complex operations that are not easily done by hand. By just entering the number and choosing the corresponding logarithmic function, the calculator quickly provides a decimal approximation.

• Features: Supports logarithmic, trigonometric, and other advanced calculations.
• User-friendly: Essential for students and professionals dealing with math regularly.

Having a scientific calculator simplifies processes by providing quick and exact results for logarithms that would otherwise be cumbersome to solve manually.

Rounding Numbers

Rounding numbers is a process to simplify a number by trimming it to a specific precision. After calculating \(\log_{10} 8.07\) using a scientific calculator, you might get a long decimal. To adhere to instructions like rounding to two decimal places, you focus on the third number after the decimal point to decide whether to round up or down.

• Precision: Rounding ensures that the number is easy to read and comprehend.
• Methods: Typically, if the third digit is five or more, you round up the second digit; otherwise, you leave it as is.

By rounding numbers, you maintain clarity and simplicity, making complex figures and calculations more manageable.