I'm not sure how to start this problem I'm able to get it to (4x-3)log(4)=(9x-6)log(3). Not sure how to continue.
Original question 4^(4x-3)=3^(9x-6)
the logs are just (messy) numbers that you can distribute, add, subtract...
[4 log(4)] x - 3 log(4) =
... [9 log(3)] x - 6 log(3)
grind away...
After a while, it's just plain algebra.
4 ^ ( 4 x - 3 ) = 3 ^ ( 9 x - 6 )
( 2 ^ 2 ) ^ ( 4 x - 3 ) = 3 ^ ( 9 x - 6 )
Take the natural logarithm of both sides and use the identity:
log ( a ^ b ) = b * log a
log 2 ^ 2 = 2 * log ( 2 )
log [ ( 2 ^ 2 ) ^ ( 4 x - 3 ) ] =
log ( 2 ^ 2 ) * ( 4 x - 3 ) =
2 * log ( 2 ) * ( 4 x - 3 )
log [ 3 ^ ( 9 x - 6 ) ]=
log ( 3 ) * ( 9 x - 6 )
So 4 ^ ( 4 x - 3 ) = 3 ^ ( 9 x - 6 ) mean:
2 * log ( 2 ) * ( 4 x - 3 ) = log ( 3 ) * ( 9 x - 6 )
2 * log ( 2 ) * 4 x - 2 * log ( 2 ) * 3 = log ( 3 ) * 9 x - log ( 3 ) * 6
8 log ( 2 ) * x - 6 log ( 2 ) = 9 log ( 3 ) * x - 6 log ( 3 ) Add 6 log ( 2 ) to both sides
8 log ( 2 ) * x - 6 log ( 2 ) + 6 log ( 2 ) = 9 log ( 3 ) * x - 6 log ( 3 ) + 6 log ( 2 )
8 log ( 2 ) * x = 9 log ( 3 ) x - 6 log ( 3 ) + 6 log ( 2 ) Subtract 9 log ( 3 ) * x to both sides
8 log ( 2 ) * x - 9 log ( 3 ) * x = 9 log ( 3 ) x - 6 log ( 3 ) + 6 log ( 2 ) - 9 log ( 3 ) * x
8 log ( 2 ) * x - 9 log ( 3 ) * x = - 6 log ( 3 ) + 6 log ( 2 )
x * [ 8 log ( 2 ) - 9 log ( 3 ) ] = 6 log ( 2 ) - 6 log ( 3 ) Divide both sides by [ 8 log ( 2 ) - 9 log ( 3 ) ]
x = 6 [ log ( 2 ) - log ( 3 ) ] / [ 8 log ( 2 ) - 9 log ( 3 ) ]
x = 0.56025
To solve the equation 4^(4x-3) = 3^(9x-6), we can use the properties of logarithms and exponentials. Here's how to continue:
Step 1: Take the logarithm of both sides
Since we have both exponential functions (4^ and 3^), taking the logarithm of both sides can help simplify the equation. You can choose any base for the logarithm, but let's use the common logarithm (base 10) or log.
log(4^(4x-3)) = log(3^(9x-6))
Step 2: Apply the exponent property of logarithms
According to the exponent property of logarithms, log(a^b) is equal to b*log(a):
(4x-3)*log(4) = (9x-6)*log(3)
Step 3: Distribute the logarithms
Distribute the logarithms on both sides of the equation to simplify further:
(4x-3)*log(4) = (9x-6)*log(3)
Step 4: Solve for x
Now you can solve the equation for x. To do that, you can divide both sides by log(4) to isolate the x term:
(4x-3)*log(4) / log(4) = (9x-6)*log(3) / log(4)
Simplifying further, you get:
4x-3 = (9x-6)*log(3) / log(4)
Now you have an equation with only x on one side. You can continue solving for x by isolating the x term and simplifying further until you obtain the final solution.