2/11/2024 0 Comments Pre calculus basicsThis also applies when the arguments are algebraic expressions. ![]() In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. ![]() To check, we can substitute x = 9 x = 9 into the original equation: log 2 ( 9 − 1 ) = log 2 ( 8 ) = 3. So, if x − 1 = 8, x − 1 = 8, then we can solve for x, x, and we get x = 9. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for x x: To solve for x, x, we use the division property of exponents to rewrite the right side so that both sides have the common base, 3. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.įor example, consider the equation 3 4 x − 7 = 3 2 x 3. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. This also applies when the exponents are algebraic expressions. In other words, when an exponential equation has the same base on each side, the exponents must be equal. Recall that the one-to-one property of exponential functions tells us that, for any real numbers b, b, S, S, and T, T, where b > 0, b ≠ 1, b > 0, b ≠ 1, b S = b T b S = b T if and only if S = T. The first technique involves two functions with like bases. Using Like Bases to Solve Exponential Equations ![]() In this section, we will learn techniques for solving exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. In fewer than ten years, the rabbit population numbered in the millions. Because Australia had few predators and ample food, the rabbit population exploded. In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. The rabbit population grew so quickly in Australia that the event became known as the “rabbit plague.” (credit: Richard Taylor, Flickr)
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