document.write( "Question 813844: please help me to solve this logarithmic :
\n" ); document.write( " Ln x = 1/2 Ln (2x + 5/2) + 1/2 Ln 2 \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Thank U.... \n" ); document.write( "

Algebra.Com's Answer #489994 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"ln%28x%29+=+%281%2F2%29ln%282x+%2B+5%2F2%29+%2B+%281%2F2%29ln%282%29\"$
\n" ); document.write( "A general procedure for solving these kinds of equations:

Use algebra and/or properties of logarithms to transform the equation into one of the following forms:
- log(expression) = other-expression
- log(expression) = log(other-expression) (Note: The bases of the two logs must match.)
Eliminate the logarithms:
- If the equation is in the first form, \"log(expression) = other-expression\", rewrite the equation in exponential form.
- If the equation is in the second form, \"log(expression) = log(other-expression)\", set the arguments equal.
Now that the logs are gone, solve the equation (using techniques which are appropriate for the type of equation it is).
Check your solution. This is not optional! A check must be made to see if the bases and arguments of all logs are valid. Any \"solution\" which make any base or an argument invalid must be rejected! (Note: Valid bases are positive but not 1 and valid arguments are positive.)

Let's try this on your equation. First we decide which form we think will be easiest to achieve. Since the terms in your equation are all logarithms, we will aim for the second, \"all-log\" form.

\n" ); document.write( "Stage 1: Transform
\n" ); document.write( "To reach this form, all we need to do is find a way to combine the terms on the right side into a single logarithm. For this we will use the $\"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"$ property. But in order for us to use this property the coefficients must be 1's, not the (1/2)'s they are now. Fortunately there is another property of logs, $\"n%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5En%29%29\"$ , which allows us to \"move\" a coefficient into the argument as its exponent. Using this property on your equation we get:
\n" ); document.write( " $\"ln%28x%29+=+ln%28%282x+%2B+5%2F2%29%5E%281%2F2%29%29+%2B+ln%282%5E%281%2F2%29%29\"$
\n" ); document.write( "Since an exponent of 1/2 means square root, I'm going to rewrite the arguments with square roots:
\n" ); document.write( " $\"ln%28x%29+=+ln%28sqrt%282x+%2B+5%2F2%29%29+%2B+ln%28sqrt%282%29%29\"$

\n" ); document.write( "Now we will use the other property to combine the two logs on the right:
\n" ); document.write( " $\"ln%28x%29+=+ln%28sqrt%282x+%2B+5%2F2%29%2Asqrt%282%29%29\"$

\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"ln%28x%29+=+ln%28sqrt%284x+%2B+5%29%29\"$
\n" ); document.write( "We have now reached form 2.

\n" ); document.write( "Stage 2: Eliminate the logs.
\n" ); document.write( "With the second form we just write an equation that says the arguments are equal:
\n" ); document.write( " $\"x+=+sqrt%284x+%2B+5%29\"$

\n" ); document.write( "Srage 3: Solve
\n" ); document.write( "Our equation is now a radical equation. To solve this we will start by squaring both sides:
\n" ); document.write( " $\"%28x%29%5E2+=+%28sqrt%284x+%2B+5%29%29%5E2\"$
\n" ); document.write( " $\"x%5E2+=+4x%2B5\"$
\n" ); document.write( "The equation is now quadratic. So we want a zero on one side. Subtracting 4x and 5 from each side we get:
\n" ); document.write( " $\"x%5E2+-+4x+-+5+=+0\"$
\n" ); document.write( "Next we factor:
\n" ); document.write( "(x-5)(x+1) = 0
\n" ); document.write( "Next the Zero Product Property:
\n" ); document.write( "x-5 = 0 or x+1 = 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "x = 5 or x = -1

\n" ); document.write( "Stage 4: Check
\n" ); document.write( "Use the original equation to check:
\n" ); document.write( " $\"ln%28x%29+=+%281%2F2%29ln%282x+%2B+5%2F2%29+%2B+%281%2F2%29ln%282%29\"$
\n" ); document.write( "Checking x = 5:
\n" ); document.write( " $\"ln%28%285%29%29+=+%281%2F2%29ln%282%285%29+%2B+5%2F2%29+%2B+%281%2F2%29ln%282%29\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"ln%285%29+=+%281%2F2%29ln%2810+%2B+5%2F2%29+%2B+%281%2F2%29ln%282%29\"$
\n" ); document.write( "At this point we can already see that the bases are e's (which is valid) and the arguments are all positive. So x = 5 checks!

\n" ); document.write( "Checking x = -1:
\n" ); document.write( " $\"ln%28%28-1%29%29+=+%281%2F2%29ln%282%28-1%29+%2B+5%2F2%29+%2B+%281%2F2%29ln%282%29\"$
\n" ); document.write( "We can already see that the first argument will be negative. This is invalid. So we must reject this \"solution\".

\n" ); document.write( "So the only solution to $\"ln%28x%29+=+%281%2F2%29ln%282x+%2B+5%2F2%29+%2B+%281%2F2%29ln%282%29\"$ is x = 5. \n" ); document.write( "

\n" );