document.write( "Question 594729: Please help me to Solve: ln(x+2)-ln(x-1)=1 \n" ); document.write( "
Algebra.Com's Answer #376964 by jsmallt9(3758)\"\" \"About 
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ln(x+2)-ln(x-1)=1
\n" ); document.write( "Solving equations like this, where the variable is in the argument (or base) of a logarithm usually starts with transforming the equation into one of the following forms:
\n" ); document.write( "log(expression) = other-expression
\n" ); document.write( "or
\n" ); document.write( "log(expression) = log(other-expression) -- where the bases of the two logs are equal.

\n" ); document.write( "With the constant term of 1 (on the right side) the \"all log\" second form will be challenging to achieve. So we are going to transform your equation into the first form.

\n" ); document.write( "The first form has a single logarithm. Your equation has two logs. Somehow we must combine the two logarithms into one. They are not like terms so we cannot simply subtract them. Fortunately there is a property will we can use:
\n" ); document.write( "\"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\"
\n" ); document.write( "Using this property on your equation we get:
\n" ); document.write( "\"ln%28%28x%2B2%29%2F%28x-1%29%29=1\"
\n" ); document.write( "We now have the equation in the first form.

\n" ); document.write( "Once the equation is in the first form, the next step is to rewrite the equation in exponential form. In general
\n" ); document.write( "\"log%28a%2C+%28p%29%29+=+q\"
\n" ); document.write( "is equivalent to
\n" ); document.write( "\"a%5Eq+=+p\"
\n" ); document.write( "Using this pattern on your equation we get:
\n" ); document.write( "\"e%5E1+=+%28x%2B2%29%2F%28x-1%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"e+=+%28x%2B2%29%2F%28x-1%29\"

\n" ); document.write( "Notice how the variable is now \"exposed\". We can now use standard algebra to solve this equation for x. First let's eliminate the fraction by multiplying each side by (x-1):
\n" ); document.write( "\"%28x-1%29%2Ae+=+%28x-1%29%2A%28%28x%2B2%29%2F%28x-1%29%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"e%2Ax-e+=+x%2B2\"

\n" ); document.write( "Next we will gather the \"x-terms\" on one side and the \"non x-terms\" on the other side of the equation. Subtracting e*x and 2 from each side we get:
\n" ); document.write( "\"-e-2+=+x+-+e%2Ax\"
\n" ); document.write( "Factoring out x on the right side:
\n" ); document.write( "\"-e-2+=+x%281+-+e%29\"
\n" ); document.write( "Then we divide bot sides by (1 - e):
\n" ); document.write( "\"%28-e-2%29%2F%281+-+e%29+=+x\"
\n" ); document.write( "And we are finished. (If you need a decimal approximation, then replace the e's with 2.71828183 (or some rounded off version of that number) and simplify.) \n" ); document.write( "
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