solve for x
2^x=0.125
5^2x=39
13^x=27
2 ^ x = 0.125
Take the logarithm to the base 2 of both sides:
log ( a ^ n ) = n * log ( a )
log( base 2 ) [ 2 ^ x ] = log( base 2 ) ( 2 ) * x = 1 * x = x
Becouse: log( base 2 ) ( 2 )
0.125 = 1 / 8 = 1 / 2 ^ 3 = 2 ^ ( - 3 )
log( base 2 ) [ 2 ^ ( - 3 ) ] = - 3
log( base 2 ) [ 2 ^ x ] = - 3
5 ^ ( 2 x ) = 39
2 x * log ( 5 ) = log ( 39 ) Divide both sides by ] 2 * log ( 5 ) ]
x = log ( 39 ) / [ 2 log ( 5 ) ]
x = 1.13815
13 ^ x = 27
x * log ( 13 ) = log ( 27 ) Divide both sides by log ( 13 )
x = log ( 27 ) / log ( 13 )
x = 1.28495
log( base 2 ) [ 2 ^ ( - 3 ) ] = - 3
x = -3
To solve these equations, you need to use logarithms. Here's how you can solve each equation step by step:
1. 2^x = 0.125:
To solve for x, we can take the logarithm of both sides using the base 2 logarithm (log base 2). The equation becomes:
log base 2 (2^x) = log base 2 (0.125)
Now, we use the logarithmic property that states log base b (b^x) = x. Applying this property, we get:
x = log base 2 (0.125)
Calculating the logarithm of 0.125 to the base 2, we get:
x = -3
So the solution to the equation 2^x = 0.125 is x = -3.
2. 5^(2x) = 39:
To solve for x, we can take the logarithm of both sides using the base 5 logarithm (log base 5). The equation becomes:
log base 5 (5^(2x)) = log base 5 (39)
Using the logarithmic property log base b (b^x) = x, we have:
2x = log base 5 (39)
Now, divide both sides of the equation by 2 to isolate x:
x = (1/2) * log base 5 (39)
Using a calculator to find the value of log base 5 (39), we get:
x ≈ 1.672
So the solution to the equation 5^(2x) = 39 is x ≈ 1.672.
3. 13^x = 27:
To solve for x, we can take the logarithm of both sides using the base 13 logarithm (log base 13). The equation becomes:
log base 13 (13^x) = log base 13 (27)
Applying the logarithmic property log base b (b^x) = x, we get:
x = log base 13 (27)
Calculating the logarithm of 27 to the base 13, we find:
x ≈ 1.471
So the solution to the equation 13^x = 27 is x ≈ 1.471.