You can put this solution on YOUR website! Solve for x: Subtract 5 from both sides. Divide by 2. Rewrite this in exponential form. or Evaluate using your calculator. Approx.
You can put this solution on YOUR website! Given to solve for x:
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Note that is the normal way to write the use of natural logarithms. It can also be written as , but this is not the usual convention. Further down I will be using instead of since it is a more general form that also can be applied to logarithms with bases other than e.
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Get the stand alone constants on the same side (the right side) by subtracting 5 from both sides as follows:
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This converts the given equation to:
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Simplify this equation by dividing both sides by 2 to get:
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Now you can solve for x by converting from the logarithmic form to its equivalent exponential form. The rule that applies is as follows:
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Given the logarithmic form:
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You can transform this to its exponential form as follows:
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This equivalence is true by the definition of a logarithm. The logarithm is defined as follows: "The logarithm of a quantity (A) is the exponent (E) that raises the base of the logarithm (B) to equal the quantity that the log is operating on (A)."
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For this problem, the value of B, the base of the logarithm, is e, the base of natural logarithms. (The value of e to 9 decimal places is 2.718281828). In the logarithmic form we have found that the right side of the equation is -1/2 and it is E. And we have that the logarithm operation is acting on x (which is A).
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By substituting those values into the exponential form, we find that:
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Transpose (reverse the sides) to get this equation into the conventional form with the unknown on the left side of the equal side and you have:
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And that is the answer to this problem in conventional form.
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Since the exponent 1/2 means the square root of and the negative sign means that it can go to a positive exponent in the denominator, you can also view this answer in the form:
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but convention is that you do not leave a radical in the denominator.
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Anyhow, you can use a scientific calculator to get a numeric answer. Either take the square root of e and divide that into 1 or use the function key and use that to raise e to the negative 1/2 power. You should get an answer of:
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or thereabouts.
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Hope this helps you to understand how to work this problem. You should become familiar with the relationship between the logarithmic and exponential forms that can be applied to problems such as this.
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