Problem 29 - Algebra 2 and Trigonometry

Question

In \(24-35,\) for each given logarithm, find the antilogarithm, \(x .\) Write the answer to four decimal places. $$ \log x=2.5619 $$

Step-by-Step Solution

Answer

The antilogarithm is approximately 363.0781.

Step 1: Understanding the Logarithm

The problem gives that \( \log x = 2.5619 \). Our task is to find the antilogarithm, which is \( x \). The antilogarithm is the inverse operation of taking a logarithm so that if \( \log_{10}x = 2.5619 \), then \( x = 10^{2.5619} \).

Step 2: Calculating the Antilogarithm

To calculate \( x = 10^{2.5619} \), we use a calculator. We input the value as a base of 10 raised to the power of 2.5619.

Step 3: Finding the Numerical Solution

Using a calculator, calculate \( 10^{2.5619} \), which gives approximately 363.0781. Ensure to check the result by calculation again for accuracy.

Step 4: Rounding to Four Decimal Places

The result 363.0781 is rounded to four decimal places, giving us 363.0781 itself as it is already at the required precision.

Key Concepts

Logarithm

Logarithms are really useful when dealing with very big or tiny numbers. They help simplify complex calculations. A logarithm answers this question: "To what power must the base be raised to produce a given number?" For example, if we say \( \log_{10} x = y \), it means "10 raised to what power gives us \( x \)?" This power is \( y \). Logarithmic operations make multiplication, division, and finding powers easier to handle.

• If \( \log_{10}(100) = 2 \), it means that 10 squared equals 100.
• Similarly, \( \log_{10}(1000) = 3 \) because 10 cubed equals 1000.

The logarithm in our exercise is \( \log_{10} x = 2.5619 \), which implies that 10 must be raised to the power of 2.5619 to get \( x \). This hints us toward finding the antilogarithm.

Inverse Operation

An inverse operation is like undoing what you've done. If tying your shoes is an operation, untying them is the inverse operation. Similarly, the antilogarithm is the inverse of the logarithm. If you know \( \log_{10} x \), you can find \( x \) by doing the opposite operation of taking the logarithm, which means raising 10 to the power of \( \log_{10} x \).

• For our exercise, the inverse operation begins with the equation \( \log_{10} x = 2.5619 \).
• To find \( x \), we must perform the inverse: \( x = 10^{2.5619} \).

This brings us to calculating the antilogarithm to find the actual number \( x \).

Numerical Solution

Finding a numerical solution involves calculating a number value for a given problem. When working through mathematical problems like our current one, calculators can be handy for this purpose. For our current exercise, we calculate the power: - Begin with the base 10
- Raise it to the power given by the logarithm: \( 10^{2.5619} \).
Using a calculator:

• Input the value 10.
• Use the exponentiation button, often represented by \( x^y \) or similar.
• Enter the power 2.5619.
• The calculator will provide the numerical solution, which is approximately 363.0781.

Computers or scientific calculators provide quick and accurate results, verifying such solutions easily.

Rounding

Rounding is crucial when you need to present a number with a certain number of digits. This often makes values easier to read and practical for use in further calculations. In mathematics, precision is key, and rounding brings this precision to an appropriate level. For our exercise:

• We calculated \( x \) and got about 363.0781.
• We needed four decimal places for precision: 363.0781 already has this.

When a number ends exactly between two rounding possibilities (e.g., 3.5 going to 4), standard rounding rules often tap up to the higher number to ensure consistency.
In the real world, correct rounding prevents errors from propagating in further calculations.