SOLUTION: 3) Write log6 (7) = x as an exponential equation. Use your calculator to evaluate x to four decimal places. __________ is the exponential form of the equation. X= __________

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: 3) Write log6 (7) = x as an exponential equation. Use your calculator to evaluate x to four decimal places. __________ is the exponential form of the equation. X= __________       Log On


   

Question 470693: 3) Write log6 (7) = x as an exponential equation. Use your calculator to evaluate x to four decimal places.
__________ is the exponential form of the equation. X= __________
Found 2 solutions by Alan3354, bucky:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Write log6 (7) = x
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6%5Ex+=+7
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Use your calculator to evaluate x to four decimal places.
x+=+log%286%2C7%29
x = log(7)/log(6)
x =~ 1.0860

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
To convert the given equation to exponential form:
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Step 1: Identify the base of the logarithm involved. It is 6
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Step 2: Then raise this base to the power on the right side of the equation. So you raise 6 to the power X.
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Step 3: Identify the term that the log function is operating on. This term is 7
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Step 4: Set the exponential term from Step 2 equal to the term in Step 3.
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The exponential equation that is equivalent to the log equation given in the problem is what you have in Step 4. It is:
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6%5Ex+=+7
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This is the answer to the first part of your problem. Next you are to solve this exponential equation by finding the value of x.
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Whenever you find a variable in an exponent, you should start thinking in terms of using a logarithm operator as a possible method of solving for the variable. In this case, it does not matter what base of the logarithm you use. However, it is a good idea to use base 10 or base e (the natural logarithm). Why? Because you can easily use a calculator to find values!
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So, to solve the exponential equation we developed ... that is to solve:
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6%5Ex+=+7
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Let's arbitrarily operate on both sides of this equation using the Log to the base 10 operator. In other words, take the log to the base 10 of both sides as follows:
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log%286%5Ex%29+=+log%287%29
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Now by the rules of logarithms, the exponent x can be brought outside the log function by making it a multiplier of the log function. When you do this, notice below how the left side of the equation is changed:
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x%2Alog%286%29+=+log%287%29
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Recognize the fact that the two log terms are just numbers. You can use a calculator to find those numbers. On a scientific calculator you can enter 6 then press the log key to find that the value of log(6) is 0.77815125. Similarly, you can find the value of log(7) by entering 7 on the calculator and then pressing the log key. The value you get for log(7) is 0.84509804. Substitute these two values into the equation and you have:
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x%2A%280.77815125%29=+0.84509804
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This is now just a straightforward algebra problem. Solve for x by dividing both sides of this equation by 0.77815125 to get:
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x+=+0.84509804%2F0.77815125
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By using your calculator to do the division on the right side you get the answer:
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x+=+1.086033133
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Just to check this answer you can return to the exponential form:
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6%5Ex+=+7
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Substitute the value you found for x:
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6%5E1.086033133+=+7
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Find the value of left side by using your calculator. Look for the x%5Ey key (or some similar key). Enter 6 on your calculator and then press the x%5Ey key. Next enter the value of the exponent 1.086033133 and finally press the = key. You should get the answer 7 for the left side and note that it does equal the 7 on the right side of the equation. Looks good!
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Hope this helps you understand a little bit more about converting logarithm terms to exponential form and how to work with logarithmic equations.