Question

In Exercises 67-70, you solved exponential and logarithmic equations with different bases. Describe general methods for solving such equations.

Short answer

There are general methods to solve both types of equations. Exponential equations can be solved by equating bases and then equating exponents, or by applying logarithms. Logarithmic equations can be solved using the change of base formula to have same base logarithms, after which comparing arguments of the logarithms is possible. Further, inverse properties of exponential and logarithmic functions can assist in the solving process.

Step-by-step solution

Solve Exponential Equations with Different Bases

Firstly, let's deal with exponential equations in which bases are different. Regarding those, attempt to make the bases the same. If this is not possible, then apply a logarithm (e.g., natural log, log base 10) to both sides of the equation to help you solve for the variable. Once you have applied the logarithm, use the properties of logs to simplify the equation and then solve.

Solve Logarithmic Equations with Different Bases

Now, to deal with logarithmic equations, we need to recall the change of base formula for logarithms which is: \(log_b{a} = \frac{log_c{a}}{log_c{b}}\) where c could be any positive value other than 1. By making use of this formula, you can change all logarithms in your equation to have the same base, which then allows you to equate the arguments of the logarithms if they are equal.

Use Inverse Properties of Exponential and Logarithmic Functions

Use the fact that exponential and logarithmic functions are inverses of each other to solve the equation. In other words, if the bases are the same, you can set the exponents equal to each other in an exponential equation. Likewise, in a logarithmic equation, if the logs (with the same base) of two quantities are equal, the quantities themselves are equal.

Key concepts

Change of Base Formula

When you're working with logarithmic equations involving different bases, a valuable tool at your disposal is the change of base formula. This mathematical gem lets you convert a logarithm with one base into an expression involving logarithms of another base, which is often more convenient to handle. For example, if you encounter a logarithm like \( \log_b{a} \), and you wish to use base 10 or the natural logarithm base (\( e \)), you can transform it using the formula:

\[ \log_b{a} = \frac{\log_c{a}}{\log_c{b}} \]
Here, \( c \) is your new base, which could be 10, \( e \), or any other positive number except 1. In practice, choosing base 10 or \( e \) makes it easier because most calculators have a direct function for these logarithms. This formula is instrumental when comparing two logarithms with unlike bases or when you must apply calculus-level operations like derivatives or integrals to logarithmic functions.

Properties of Logarithms

Understanding the properties of logarithms can greatly simplify the process of solving logarithmic equations. These properties emerge from the definition of a logarithm as the inverse of an exponential function. Here are a few key properties that are especially handy:

• Product Property: The logarithm of a product is equal to the sum of the logarithms of the factors, represented by \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• Quotient Property: The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator, which looks like \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \).
• Power Property: The logarithm of a number raised to a power is that power times the logarithm of the number, exemplified by \( \log_b(m^n) = n\cdot\log_b(m) \).

By applying these properties, you can often rewrite complex logarithmic expressions in a way that makes variables easier to isolate and solve for.

Inverse Properties of Exponential and Logarithmic Functions

Exponential and logarithmic functions are best buddies because they are inverses of each other. This relationship is paramount when you're trying to solve equations involving these functions. An exponential function of the form \( b^x = y \) has an inverse logarithmic function \( \log_b{y} = x \), and vice versa.

This inverse property means that if you take the logarithm of both sides of an exponential equation, the exponent on one side can be brought down in front of the logarithm, turning it into a linear form, which is easier to solve. Similarly, if you have a logarithmic equation and you exponentiate both sides using the base of the logarithm, the logarithmic operation and exponential operation will cancel each other out, leaving you with a simpler equation.

Simplify Equations

The goal when solving any equation is to isolate the variable you're solving for. Simplification is vital in reducing complex expressions down to something manageable. When it comes to simplifying equations, this often involves applying the distributive property, combining like terms, and using methods to move terms from one side of the equation to the other without changing the equation's balance.

In the case of exponential and logarithmic equations, simplification might include using the inverse properties to get rid of the exponential or logarithmic part altogether. You may also use logarithmic properties to combine multiple logarithmic terms into a single one or to break down a complicated logarithmic term into several simpler ones. Keeping equations simple and manageable is the key to a clear path to the solution.