I have to solve the equation by finding the exact solution. Please Help.
1/4 log(x)-log(10000)=4
lgt ( x ) = The base 10 logaritam
lgt ( 10000 ) = 4
( 1/4 ) lgt (x) - lgt( 10000 ) = 4
( 1/4 ) lgt (x) - 4 = 4
( 1/4 ) lgt (x) = 4 + 4
( 1/4 ) lgt (x) = 8 Multiply both sides with 4
lgt ( x ) = 32
10 ^ lgt ( x ) = 10 ^ 32
Remark : 10 ^ lgt ( x ) = x
x = 10 ^ 32
To solve the equation 1/4 log(x) - log(10000) = 4 and find the exact solution, we'll go through the steps together:
Step 1: Combine the logarithmic terms on the left side of the equation.
Simplifying the equation, we have: log(x^(1/4)) - log(10000) = 4.
Step 2: Use the properties of logarithms to rewrite the equation.
Applying the quotient rule of logarithms, we can rewrite the equation as:
log(x^(1/4) / 10000) = 4.
Step 3: Rewrite 10000 as a power of 10.
10000 can be written as 10^4 since it's equal to 10,000.
Step 4: Simplify the equation further.
Now we have log(x^(1/4) / 10^4) = 4. We can simplify x^(1/4) / 10^4 as x^(1/4) / (10^4), giving us:
log(x^(1/4)) - log(10^4) = 4.
Step 5: Apply the power rule of logarithms to simplify further.
According to the power rule, log(x^n) = n * log(x). Applying this to the equation, we have:
(1/4) * log(x) - 4 * log(10) = 4.
Step 6: Calculate the value of log(10).
Logarithm of base 10 for the number 10 is equal to 1. So log(10) = 1.
Step 7: Substitute the value of log(10) into the equation.
The equation now becomes:
(1/4) * log(x) - 4 * (1) = 4.
Simplifying further, we get:
(1/4) * log(x) - 4 = 4.
Step 8: Move the constant term to the other side of the equation.
By adding 4 to both sides of the equation, we get:
(1/4) * log(x) = 8.
Step 9: Isolate the logarithmic term.
To isolate the logarithmic term, we multiply both sides of the equation by 4:
4 * (1/4) * log(x) = 4 * 8,
which simplifies to:
log(x) = 32.
Step 10: Convert the logarithmic equation into exponential form.
In exponential form, log(x) = y is equivalent to x = base^y. Therefore, the equation becomes:
x = 10^32.
The exact solution to the equation 1/4 log(x) - log(10000) = 4 is x = 10^32.