document.write( "Question 614549: Solve for x, to 2 decimal places: $\"+6%5E%283x%29+=+4%5E%282%2Ax-3%29+\"$ \n" ); document.write( "

Algebra.Com's Answer #386698 by jsmallt9(3758) $\"\"$ $\"About$

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$\"+6%5E%283x%29+=+4%5E%282%2Ax-3%29+\"$
\n" ); document.write( "Solving an equation where the variable is in an exponent usually starts with isolating the base and its exponent. In this equation we have two bases and exponents. But they are both already isolated.

\n" ); document.write( "The next step is to use logarithms. Any base of logarithm can be used but there are advantages to choosing certain bases:

Choosing a base for the logarithm that matches the base of the exponent will result in the simplest expression for the answer.
Choosing a base for the logarithm that your calculator \"knows\", base 10 or base e (aka ln), will result in a less simple expression but one that can be easily converted into a decimal approximation.

Since there are two different bases we cannot match them with the base of a single logarithm. And even if we could match bases, we have been told to find a decimal answer. So we will use base 10 or base e logarithms. Either will work. I'll choose ln. Finding the ln of each side:
\n" ); document.write( " $\"+ln%286%5E%283x%29%29+=+ln%284%5E%282%2Ax-3%29%29+\"$

\n" ); document.write( "Next we use a property of exponents, $\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"$ , which allows us to move the exponent of the argument of a logarithm out in front of the logarithm. (This property is the very reason we use logarithms on equations like this. Being able to move an exponent, where the variable is, out in front gives us an equation we can now solve. This property applies to all bases of logarithms which is why any base of logarithm can be used.) Using this property on both logarithms we get:
\n" ); document.write( " $\"+%283x%29ln%286%29+=+%282%2Ax-3%29ln%284%29+\"$

\n" ); document.write( "Now that the variable is finally out of the exponents, we can solve for x. First we simplify. Using the distributive property on the right side:
\n" ); document.write( " $\"+%283x%29ln%286%29+=+%282x%29ln%284%29+-+3ln%284%29+\"$
\n" ); document.write( "Gather the x terms on one side. Subtracting (2x)ln(4) from each side:
\n" ); document.write( " $\"+%283x%29ln%286%29+-+%282x%29ln%284%29+=+-3ln%284%29+\"$
\n" ); document.write( "The terms on the left are not like terms so we cannot subtract them. But we can factor out x:
\n" ); document.write( " $\"+x%283ln%286%29+-+2ln%284%29%29+=+-3ln%284%29+\"$
\n" ); document.write( "Now we divide both sides by ((3ln(6) - 2ln(4))\"
\n" ); document.write( " $\"+x+=+%28-3ln%284%29%29%2F%283ln%286%29+-+2ln%284%29%29+\"$
\n" ); document.write( "This is an exact expression for the solution to your equation. I'll leave it up to you and your calculator to get a decimal approximation and round it.

\n" ); document.write( "Note 1: If your calculator has buttons for parentheses, you could type in the whoie thing:
\n" ); document.write( "(-3ln(4))/((3ln(6) - 2ln(4))
\n" ); document.write( "Note 2: If your calculator does not have parentheses buttons, then be sure to find the logarithms before you do any multiplying, dividing or subtracting. \n" ); document.write( "

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