Exponential and logarithmic functions in carbon 14 dating when dating someone whos separated
\(2^=2^\) To take a power of a power, multiply exponents.\(3x 6=4x 4\) Use the one-to-one property to set the exponents equal. Example \(\Page Index\): Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base Solve \(2^=\sqrt\). Sometimes the terms of an exponential equation cannot be rewritten with a common base.We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression. Figure \(\Page Index\) represents the graph of the equation.For example, consider the equation \(_2(2) _2(3x−5)=3\). On the graph, the 0\) and any positive real number \(b\), where \(b≠1\), \(_b S=_b T\) if and only if \(S=T\). So, if \(x−1=8\), then we can solve for \(x\),and we get \(x=9\).
\(8=2x−10\) Apply the one-to-one property of exponents. Solution: \(8^=^\) \(^=^\) Write \(8\) and \(16\) as powers of \(2\).
\(e^−e^x=56\) \(e^−e^x−56=0\) Get one side of the equation equal to zero. \(e^x 7=0\) or \(e^x−8=0\) If a product is zero, then one factor must be zero. \(e^x=8\) Reject the equation in which the power equals a negative number.
\(x=\ln8\) Solve the equation in which the power equals a positive number.
Solution: \(2^=2^\) Write the square root of \(2\) as a power of \(2\). In these cases, we solve by taking the logarithm of each side.
Recall, since \(\log(a)=\log(b)\) is equivalent to \(a=b\), we may apply logarithms with the same base on both sides of an exponential equation. \(x\ln5−x\ln4=−2\ln5\) Get terms containing \(x\) on one side, terms without \(x\) on the other. One common type of exponential equations are those with base \(e\).