2.13 Exponential and logarithmic equations and inequalities

Certain exponential equations arise that can be easily solved without resorting to logarithms. The key to their solution lies in the recognition of various powers of integers. We can solve these types of equations by rewriting both sides of the equation with the same base to some power. If we can do this, we can use the equality property of exponents which says b^x=b^y \iff x=y If both sides of the equation cannot be written in the same base, another method of solving exponential equations is taking the logarithm of both sides. Then, we can use properties of logarithms to solve the equation. Often, we will evaluate the solutions to these equations with a calculator.

If these methods do not work, we can use technology to estimate solutions. To solve with a graphing calculator, we can graph the expression on the left side of the equation, graph the expression on the right side of the equation, and then find their point of intersection.

Examples

Many logarithmic equations can be solved by rewriting it in exponential form. The properties of logarithms may need to be used to condense the equation down to one logarithm before switching it to exponential form.

Recall a logarithmic function of the form \log_b\left(x\right) has domain x>0. When solving logarithmic equations, we need to ensure our answers make sense in context and that we are never taking the logarithm of a negative number. We may find that, for some logarithmic equations, a solution resulting from the process of solving is extraneous because it results in a negative argument.

Logarithmic equations which have non-zero expressions on both sides can be solved graphically. To do this, set the expressions on both sides equal to y, graph each equation on the same coordinate plane, then find the point(s) of intersection.

Examples

Exponential inequalities are inequalities that include an exponent. These inequalities come in various forms, such as b^{x} \gt k,\, b^{x} \lt k,\, b^{x} \geq k, or b^{x}\leq k, where b is the base, x is the exponent, and k is a constant. The base, b, is a positive number not equal to 1. Just like exponential equations, in an exponential inequality, the base of the exponential expression can be a number, a variable, or even a more complicated expression. It's also important to note that exponential inequalities, like exponential equations, deal with exponential growth or decay situations.

Solving exponential inequalities often requires a good understanding of both exponential functions and logarithmic functions. That's because logarithms are the inverses of exponential functions, and they provide us with the tools we need to solve for variables that are in the exponent of an exponential inequality.

To solve an exponential inequality using logarithms, we generally start by isolating the exponential expression on one side of the inequality. Then, we can apply a logarithm to both sides of the inequality. This allows us to "bring down" the exponent and turn it into a coefficient, which is often easier to work with.

The graph above shows an exponential function, y = 2^{x}, and its corresponding logarithmic function, y = \text{log}_{2}x. You'll notice that these functions are reflections of each other across the line y = x. This illustrates the idea that logarithms "undo" or "reverse" the operation of exponentiation, which is crucial for understanding how to solve exponential inequalities.

Remember, when dealing with inequalities, it's important to keep in mind that the direction of the inequality sign may change when both sides are multiplied or divided by a negative number or when taking the logarithm of both sides, as \text{log}_{b}a \gt \text{log}_{b}c if and only if a \gt c for b \gt 1.

Examples

Much like exponential inequalities, logarithmic inequalities involve a comparison of two logarithmic expressions or a logarithmic expression and a number. The main difference is that, in this case, the variable we're solving for is in the argument of the logarithm, not the exponent. For example, an inequality like \text{log}_\text{base} b (x) \gt a.

Remember the important properties of logarithms that you've learned before. The logarithm is the inverse operation of exponentiation, which means that they undo each other. Also, logarithms have a key property that \text{log} (a \times b) = \text{log}(a) + \text{log}(b), and \text{log}\left(\dfrac{a}{b}\right) = \text{log}(a) - \text{log}(b). These properties are helpful when solving logarithmic inequalities.

Solving logarithmic inequalities generally involves using properties of logarithms, converting to exponential form, and careful consideration of the domain of the logarithm.

Remember that the domain of a logarithm is all positive real numbers. That is, we can only take the logarithm of a positive number. This is crucial to remember when solving logarithmic inequalities because it affects the solutions of the inequality.

Moreover, when solving inequalities involving logarithms, the direction of the inequality symbol depends on the base of the logarithm:

• If the base is greater than 1 (such as in \text{log}_2(x) or \text{log}_10(x)), the inequality symbol remains the same when you convert the inequality from logarithmic form to exponential form.
• If the base is between 0 and 1 (like in \text{log}_{0.5}(x) or \text{log}_\frac{1}{3}(x)), the inequality symbol flips direction when you convert the inequality from logarithmic form to exponential form.

Examples