Problem 10 - Technical Mathematics with Calculus

Question

Find the value of \(x\) in each expression. $$\log _{3} x=4$$

Step-by-Step Solution

Answer

The value of \(x\) is 81.

Step 1: Understand the Given Equation

The given equation is in logarithmic form. It states that the logarithm of x to the base 3 is equal to 4. In mathematical terms, this can be written as \(\log _{3} x=4\).

Step 2: Convert from Logarithmic to Exponential Form

To find the value of x, we need to convert the logarithmic equation into an exponential form. This is done by using the definition of a logarithm which states that if \(\log _{b} a=c\), then \(b^{c}=a\). Using this definition, we can rewrite \(\log _{3} x=4\) as \(3^{4}=x\).

Step 3: Calculate the Value of x

After rewriting the equation in exponential form, we can calculate the value of x by raising 3 to the power of 4. The result is \(3^{4}=3\times3\times3\times3=81\), so \(x=81\).

Key Concepts

Converting Logarithmic to Exponential Form

When working with logarithms, understanding how to transform them into exponential expressions is crucial for solving equations. A logarithm basically answers the question: 'To what power must the base be raised to yield a certain number?' For example, in the equation \(\log_{3}x=4\), we're searching for the power to which 3 must be raised to get the value of \(x\).

Using the rule \(\log_{b}a=c \Rightarrow b^{c}=a\), you can convert any logarithmic equation to an exponential form. In the exercise provided, \(\log_{3}x=4\) becomes \(3^{4}=x\). This approach simplifies the process as you're no longer dealing with a logarithm but a straightforward multiplication.

After converting, you move on to the actual computation or 'exponential equations' phase, which brings us to our next topic.

Exponential Equations

An exponential equation is one where the unknown variable appears as an exponent. In our original problem, once converted, we have \(3^{4}=x\), which is an exponential equation. Not all exponential equations are as simple to compute as this one, but the basic process involves isolating the variable and then finding its value through calculation or, in more complex cases, using logarithms.

In the provided step-by-step solution, the calculation is straightforward: \(3^{4}\) represents 3 multiplied by itself 4 times, resulting in 81. Remember, the base \(3\) here is being raised to the fourth power \(4\), which forms the core of understanding exponential relationships between numbers. It's like asking, 'How many times does the factor 3 appear in the product when repeated 4 times?' This mental image can often simplify the comprehension of exponential growth.

Solving Logarithmic Equations

Finally, let's talk about the broader topic of 'solving logarithmic equations.' These are equations where the unknown variable is part of a logarithm. Solving them requires understanding both logarithmic and exponential forms because, often, you'll need to switch between the two to isolate and solve for the variable.

The key to solving logarithmic equations is transforming them into a form where the variable is easier to isolate - typically converting them into exponential form as demonstrated in our exercise. However, not all logarithmic equations are as direct, especially when they involve multiple logarithmic terms or different bases. In such cases, you may need to employ properties of logarithms, such as product, quotient, and power rules, to combine or simplify them before conversion.

Understanding these core concepts - converting forms, exponential calculations, and solving logarithmic steps - builds a strong foundation for tackling a variety of mathematical challenges related to exponential and logarithmic equations.