Problem 14 - Intermediate Algebra

Question

For Problems \(11-20\), use your calculator to find \(x\) when given \(\log x\). Express answers to five significant digits. $$ \log x=3.9335 $$

Step-by-Step Solution

Answer

x ≈ 8560.5

Step 1: Understanding Logarithms

A logarithm tells us the power to which a number must be raised to obtain another number. In this problem, \(\log x = 3.9335\) implies that \(x\) is equal to 10 raised to the power of 3.9335, since base 10 is implied in the notation.

Step 2: Convert Logarithmic Equation to Exponential Form

To find the value of \(x\), we need to convert the logarithmic equation into exponential form. This can be done as follows: \(x = 10^{3.9335}\).

Step 3: Use Calculator to Compute the Exponential

Enter the expression \(10^{3.9335}\) into a calculator. Make sure your calculator is set to use the common logarithm (base 10).This will give the value of \(x\) as approximately 8560.499.

Key Concepts

Exponential Form

When dealing with logarithms, it's important to understand exponential form. Essentially, when you see a logarithmic equation like \(\log x = 3.9335\), this implies an exponential relationship. The base of the logarithm, which is commonly 10 in these exercises, plays a critical role.
In general, if you have \(\log_{b}a = c\), this tells you that \(b^c = a\). In our example \(\log x = 3.9335\), the base is 10 implicitly, as \(\log\) without a base is common log (base 10). So, to convert:\

• Rewrite \(\log x = 3.9335\) as \(10^{3.9335} = x\).
• This shows how the logarithmic equation becomes an exponential equation.

Exponential form gives us a straightforward path to finding \(x\) using simple calculations. It's a translation from one form (logarithmic) to another (exponential) that enhances our ability to find answers quickly.

Finding Values Using Calculators

Once you have the exponential form \(x = 10^{3.9335}\), you'll need a calculator to find the exact value of \(x\). Calculating powers manually can be daunting and impractical, thus making a calculator essential.
To find the value using a calculator:

• Ensure your calculator is in the correct mode. For common logarithms, this is base 10.
• Enter the number 10, raised to the power of 3.9335, which means typing in something like \(10^{3.9335}\).
• After computing, most calculators will directly give you the result: approximately 8560.499.

It's crucial that the calculator settings are correct, particularly ensuring it's calculating powers using a base of 10. This helps avoid errors in answers and confirms the accuracy of your computations.

Common Logarithms

Common logarithms are logarithms with a base of 10. They are frequently used in sciences and engineering because they are simpler to handle due to the base 10 numbering system we use daily.
Common logarithms are often written simply as \(\log\) without noting the base. So when you see \(\log x\), it means \(\log_{10}x\). This can lead to simpler calculations when used with a calculator.
Key Points about Common Logarithms:

• Common logarithms relate directly to powers of 10, which means every increment represents a tenfold increase.
• They are particularly useful for simplifying multiplicative relationships into additive ones.
• Understanding \(\log x = c\) implies \(x = 10^c\) is fundamental.

Grasping the concept of common logarithms allows us to manipulate and solve equations with logarithmic expressions efficiently, particularly when finding powers of 10 using calculators.