Question

How do you solve the following exponential equation by expressing each power of the same base and then equating the exponents?

Answer 1

$6^{x-\tfrac{3}{4}}=\sqrt{6}\\ 6^{x-\tfrac{3}{4}}=6^{\tfrac{1}{2}}\\ x-\dfrac{3}{4}=\dfrac{1}{2}\\ x=\dfrac{1}{2}+\dfrac{3}{4}\\ x=\dfrac{2}{4}+\dfrac{3}{4}\\ x=\dfrac{5}{4}$

Answer 2

Start by switching everything into exponent form: 6^(x-3/4)=6^(1/2). In this equation, we're trying to find the value of x that will make each side equal. Because the base (6) is the same on either side of the equal sign, all we need to worry about is the exponents, because this equation can only be true if the exponents are the same (i.e. 6^(1/2)=6^(1/2)). Thus, you can actually ignore the 6's entirely and only focus on the exponents: x-3/4=1/2. Then, this has just become a simple algebraic expression. x-3/4+3/4=1/2+3/4. x=1/2(2/2)+3/4. x=2/4+3/4. x=5/4