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$35-44$ Logarithmic Equations Use the definition of the logarithmic function to find $x .$$${ (a)} \log _{x} 1000=3 \quad \text { (b) } \log _{x} 25=2$$
a) $x=10 \quad, \quad$ b) $x=5$
Algebra
Chapter 4
Exponential and Logarithmic Functions
Section 3
Logarithmic Functions
Missouri State University
Harvey Mudd College
University of Michigan - Ann Arbor
Idaho State University
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before we onto this question. Let's just recap on the definition off the logarithmic function. So if I have locked to the base off a be equal to see to rewrite this into exponential for I have the base off a to the power off, see equal to be right. So remember the base off the log becomes the base off the exponential form and then the sea becomes the exponents and the B is on the other side of the equal sign. Another thing is that you will remember that Lin off are you? Is the lock with base e off? I great. Let's see how we apply this to the question lock by speaks 1000 equal to three and we have to solve for X. So in this case, X will be the base. So it becomes it's the base of the lock. So it becomes the base off the exponents off the exponential form. Right? So and then the three is the exponents and that all is equal to 1000. But the 1000 I can rewrite a stain to the power of three. So now, because the exponents are the same, I can take the cube root on both sides on it will simplify two X equal to 10. We, we have look to the bites off aches 25 equal to two exist the base writer. He becomes the base off the exponential form. The two is the exponents, and that is all equal to 25. Now the 25 can become five squid. If I take the square root year, remember, when you take an even route, you have to add plus or minus right because plus five squared gives you 25 as well as negative. Five. Squared will also give you 25. So therefore X in this case is plus or minus negative five. But in a look, you're not allowed to have a negative value. So for that reason, we can say that X equal to negative five is not applicable in this guy's. So they fall X is only equal to five
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