Understanding the Logarithmic Function Problem

The problem asks us to evaluate a specific expression involving a logarithmic function defined as \(f(x) = \log_{10}(1 + x)\). The expression we need to calculate is \(4f(4) + 5f(1) – \log_{10} 2\). To solve this, we first need to find the values of \(f(4)\) and \(f(1)\) by substituting the respective values of \(x\) into the function \(f(x)\).

Calculating \(f(4)\) and \(f(1)\)

Let's find the values of the function \(f(x)\) at the given points, \(x=4\) and \(x=1\).

For \(x=4\): Substitute \(x=4\) into \(f(x)\). \(f(4) = \log_{10}(1 + 4) = \log_{10} 5\)

For \(x=1\): Substitute \(x=1\) into \(f(x)\). \(f(1) = \log_{10}(1 + 1) = \log_{10} 2\)

Now we have the values for \(f(4)\) and \(f(1)\) in terms of \(\log_{10}\).

Substituting Values into the Expression

The expression we need to evaluate is \(4f(4) + 5f(1) – \log_{10} 2\). Let's substitute the values we found for \(f(4)\) and \(f(1)\):

Expression = \(4(\log_{10} 5) + 5(\log_{10} 2) – \log_{10} 2\)

Simplifying the Expression Using Logarithm Properties

Now, we simplify the expression using the properties of logarithms. The expression is \(4 \log_{10} 5 + 5 \log_{10} 2 – \log_{10} 2\). First, combine the terms involving \(\log_{10} 2\):

\(5 \log_{10} 2 – \log_{10} 2 = (5-1) \log_{10} 2 = 4 \log_{10} 2\)

So, the expression becomes:

Expression = \(4 \log_{10} 5 + 4 \log_{10} 2\)

Next, we can use the logarithm property \(a \log_b c = \log_b (c^a)\) to rewrite the terms:

\(4 \log_{10} 5 = \log_{10} (5^4) = \log_{10} 625\)

\(4 \log_{10} 2 = \log_{10} (2^4) = \log_{10} 16\)

The expression is now:

Expression = \(\log_{10} 625 + \log_{10} 16\)

Now, we use the logarithm property \(\log_b X + \log_b Y = \log_b (X \cdot Y)\) to combine the terms:

Expression = \(\log_{10} (625 \times 16)\)

Calculating the Final Value

We need to calculate the product \(625 \times 16\).

\(625 \times 16 = 10000\)

So the expression is:

Expression = \(\log_{10} 10000\)

Finally, we evaluate \(\log_{10} 10000\). The base-10 logarithm of a number is the power to which 10 must be raised to get that number. Since \(10^4 = 10000\), \(\log_{10} 10000 = 4\).

The value of the expression \(4f(4) + 5f(1) – \log_{10} 2\) is 4.

Summary of Steps

Here is a quick summary of the steps taken:

Calculate \(f(4)\) and \(f(1)\) using the function definition \(f(x) = \log_{10}(1 + x)\).

Substitute these values into the given expression \(4f(4) + 5f(1) – \log_{10} 2\).

Simplify the expression by combining like terms involving logarithms.

Use logarithm properties (\(a \log_b c = \log_b (c^a)\) and \(\log_b X + \log_b Y = \log_b (X \cdot Y)\)) to further simplify.

Calculate the final logarithmic value.

Revision Table: Key Logarithm Properties

Additional Information: Understanding Logarithms and Functions

A logarithm is the inverse operation to exponentiation. This means the logarithm of a number \(X\) with respect to a base \(b\) is the exponent to which \(b\) must be raised to produce \(X\). It is written as \(\log_b X\). In this problem, we are working with the common logarithm, which has a base of 10. It is often written as \(\log_{10} X\) or simply \(\log X\).

A function, like \(f(x) = \log_{10}(1 + x)\), is a rule that assigns a unique output value (in this case, a logarithm) for each valid input value (\(x\)). To evaluate a function at a specific point, you substitute that point's value for the variable (here, \(x\)) in the function's definition and compute the result.

Solving this problem requires a good understanding of function evaluation and the fundamental properties of logarithms, particularly the power rule and the product rule.