Problem 34 - Contemporary Precalculus

Question

(a) Solve \(7^{x}=3,\) using natural logarithms. Leave your answer in logarithmic form; don't approximate with a calculator. (b) Solve \(7^{x}=3,\) using common (base 10 ) logarithms. Leave your answer in logarithmic form. (c) Use the change of base formula in Special Topics \(5.4 . \mathrm{A}\) to show that your answers in parts (a) and (b) are the same.

Step-by-Step Solution

Answer

Question: Solve the equation \(7^x = 3\) using (a) natural logarithms, (b) common logarithms, and (c) show that the answers obtained in parts (a) and (b) are the same. Answer: Using both natural and common logarithms, we find the value of x to be \(x = \frac{\ln(3)}{\ln(7)} = \frac{\log(3)}{\log(7)}\), which results in the same answer.

Step 1: Solve using natural logarithms

First, we will take the natural logarithm of both sides of the equation. The equation is given by: \(7^{x}=3\). Taking the natural logarithm: \(\ln(7^{x}) = \ln(3)\)

Step 2: Applying logarithmic identities

Next, we can use the logarithmic identity \(\ln(a^b) = b\ln a\). Applying this identity to our equation, we get: \(x\ln(7) = \ln(3)\)

Step 3: Solve for x

To solve for x, we need to isolate x on one side of the equation. We can do this by dividing both sides by \(\ln(7)\): \(x = \frac{\ln(3)}{\ln(7)}\) This is our final answer for the value of x using natural logarithms.

Step 4: Solve using common logarithms

Now we will solve the equation using common logarithms (base 10 logarithms). The equation is given by: \(7^{x}=3\). Taking the common logarithm: \(\log(7^{x}) = \log(3)\)

Step 5: Applying logarithmic identities

Again, we use the logarithmic identity \(\log(a^b) = b\log a\). Applying this identity to our equation, we get: \(x\log(7) = \log(3)\)

Step 6: Solve for x

Once again, to solve for x, we need to isolate x on one side of the equation. We can do this by dividing both sides by \(\log(7)\): \(x = \frac{\log(3)}{\log(7)}\) This is our final answer for the value of x using common logarithms.

Step 7: Use the change of base formula

Now we have to use the change of base formula to show that the answers obtained using natural logarithms and common logarithms are the same. The change of base formula states that: \(\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\) Comparing the natural logarithm solution and the common logarithm solution: \(\frac{\ln(3)}{\ln(7)} = \frac{\log(3)}{\log(7)}\) Using the change of base formula, we can rewrite the natural logarithm solution as: \(\frac{\ln(3)}{\ln(7)} = \frac{\log_e(3)}{\log_e(7)}\) And similarly, we can rewrite the common logarithm solution as: \(\frac{\log(3)}{\log(7)} = \frac{\log_{10}(3)}{\log_{10}(7)}\) By comparing these equations, we conclude that both the natural logarithm and the common logarithm solutions are the same, and the value of x is: \(x = \frac{\ln(3)}{\ln(7)} = \frac{\log(3)}{\log(7)}\)

Key Concepts

Logarithms

Logarithms are mathematical expressions used to represent power or exponents. They reverse the process of exponentiation just like division reverses multiplication. The logarithm of a number is the exponent to which a base, usually 10 or e (a mathematical constant approximately equal to 2.718), must be raised to produce that number. For example, in the expression \(\:x = \log_b{y}\), \(b\) is the base, \(y\) is the argument or number you're taking a logarithm of, and \(x\) represents the power or exponent.

• In simpler terms, if \(b^x = y\), then \(\log_b{y} = x\).
• Logarithms have special properties, such as \(\log_b{(mn)} = \log_b{m} + \log_b{n}\), which help simplify calculations.

Logarithms are used across scientific fields for calculations involving large numbers, exponential growth, and in problem-solving, as shown in the exercise solution steps.

Natural Logarithms

Natural logarithms are logarithms that have the constant e as their base. The notation for a natural logarithm is \(\ln(x)\), which reads 'lawn of x'. Natural logarithms are ubiquitous in mathematics, particularly in calculus and complex analysis.

• Unlike common logarithms which use 10 as the base, natural logarithms use e (approximately 2.718281828...).
• They are particularly useful in mathematics involving continuous growth or decay, such as population models or radioactive decay.

In the example, to solve \(7^x = 3\) using natural logarithms, we transform the equation to \(x \ln(7) = \ln(3)\). By solving, we find \(x = \frac{\ln(3)}{\ln(7)}\). This solution format helps visualize the relationship between the exponent and the logarithmic functions.

Common Logarithms

Common logarithms, often simply called 'logs', use 10 as their base. They are denoted as \(\log(x)\). Common logarithms are frequently used in engineering, science, and other fields prone to measurements spanning orders of magnitude due to their straightforward base.

• Common logarithms simplify powers of ten, making them particularly convenient for calculations in decimal systems.
• For instance, \(\log(1000) = 3\) because 10 raised to the power of 3 is 1000.

In our problem, using common logs, the expression \(7^x = 3\) transforms to \(x \log(7) = \log(3)\), leading to \(x = \frac{\log(3)}{\log(7)}\). This illustrates the utility of common logarithms in various scientific computations.

Change of Base Formula

The Change of Base formula is a useful tool for evaluating logarithms that are not in convenient bases, like 10 or e. The formula itself is: \(\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\), where \(a\) is the original base and \(c\) the new base, frequently chosen as either 10 or e for simplicity.

• This formula is particularly advantageous when dealing with logarithms on calculators, which typically support only base 10 and base e.
• It allows you to convert any logarithmic expression into another that is computable with standard calculator functions.

In the exercise, both solutions, \(\frac{\ln(3)}{\ln(7)}\) using natural logs and \(\frac{\log(3)}{\log(7)}\) using common logs, are shown to be equivalent due to this formula. It demonstrates that while we may use different logarithmic bases, the values remain consistent and interchangeable using this transformation.