SOLUTION: 5^(3y-4)=685 y=?

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Question 553685: 5^(3y-4)=685
y=?
Found 2 solutions by rapaljer, bucky:
Answer by rapaljer(4671) (Show Source):
You can put this solution on YOUR website!

Take the ln of each side:

Use the distributive property:

Add +4*ln 5 to each side:

Divide both sides by 3*ln 5:

With a calculator, I get y= 2.68565 approximately.

For a NON-TRADITIONAL explanation of this topic that is probably easier to understand than your own textbook, please see my own website! The easiest way to find it is to use the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the very bottom of this page there is a link that will take you to the Homepage of my website. I have a complete ALGEBRA curriculum there with LOTS of practice tests, and even a few videos. Best of all, it's all FREE!!!

For this particular topic, when you find the Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time", choose "College Algebra" and look in "Chapter 4" for "Section 4.04 Solving Exponential and Logarithmic Equations." There is a complete explanation, illustrated with the TI84, together with lots of examples and exercises with ALL the answers given. You will especially like the "Math in Living Color" pages that go with this, where hundreds of the hardest exercises are solved for you IN COLOR! I even have TWO videos of me teaching LOGARITHMS from before I retired. Remember, it's all FREE!

If you or anyone needs to contact me, my Email address is rapaljer@seminolestate.edu.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus

Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
You were asked to solve for y in the equation:
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Whenever you see an unknown in the exponent, you should think that taking a logarithm of the term is probably the way to solve for the unknown.
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This being the case, let's take the logarithm of the term on the left side. But whatever we do to the left side, we must also do to the right side in order to maintain the equality. So we'll take the log of both sides. What base should we use for the logs? Doesn't make any difference as long as we use the same based logs for both sides of the equation. However, from a practical standpoint (because we can use a calculator to easily find these logarithms) we should use either natural logarithms or base 10 logarithms. Let's use base 10. When we take the log of both sides we get:
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A general property of logarithms is that when you take the logarithm of a quantity that has an exponent, you can bring the exponent out as a multiplier of the logarithm. In this problem, on the left side you can bring the (3y-4) out to multiply the log and then equation then becomes:
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Now divide both the left and right sides by log(5). On the left side you are left with just 3y - 4 and the equation is then:
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On the right side the log(685) is just a number we can find with a calculator, and the same thing can be said for log(5). (Remember that we are working in base 10 logarithms.) Enter 685 on your calculator and press the log key to find that log(685) = 2.835056102. Then enter 5 and press the log key to get that log(5) = 0.698970004. By substituting these values the equation becomes:
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Doing the division on the right side reduces the equation to:
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Then get rid of the -4 on the left side by adding +4 to both sides to get:
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Finally divide both sides by 3 to get the answer:
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Hope this helps you to understand how logarithms can help you to solve for unknowns when they appear in an exponent.
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