1-log5=1/3(log1/2+log x+1/3log5)
To solve the equation 1 - log5 = 1/3(log1/2 + log x + 1/3log5), we need to apply some logarithmic properties and algebraic manipulations:
Step 1: Simplify the equation using logarithmic properties.
Start by applying the rule that states, log(a) - log(b) = log(a/b).
1 - log5 = 1/3(log1/2 + log x + 1/3log5)
1 - log5 = log((1/2 * x * 5^(1/3))/5)
Step 2: Simplify the expression inside the logarithm.
Using the property log(a) + log(b) = log(a * b), we can simplify the expression inside the logarithm:
(1/2 * x * 5^(1/3))/5 = x * (5^(1/3))/10
Substituting this simplified expression back into the equation, we have:
1 - log5 = log(x * (5^(1/3))/10)
Step 3: Simplify further by using the logarithmic property log(a^b) = b * log(a):
1 - log5 = log(x * (5^(1/3))/10)
1 - log5 = log(x * (5^(1/3))) - log(10)
1 - log5 = log(x) + log(5^(1/3)) - log(10)
Step 4: Apply the rules of logarithms to simplify.
Using the properties log(a) + log(b) = log(a * b) and log(a) - log(b) = log(a/b), we can simplify further:
1 - log5 = log(x) + log(5^(1/3)) - log(10)
1 - log5 = log(x) + 1/3log(5) - log(10)
Step 5: Combine the logarithms on the right side of the equation.
Combining the two logarithms on the right side of the equation:
1 - log5 = log(x) + 1/3log(5) - log(10)
1 - log5 = log(x) + log(5^(1/3)/10)
Step 6: Apply the logarithmic property to convert the equation into exponential form.
Using the property log(a) = b is equivalent to a = 10^b, we can rewrite the equation:
1 - log5 = log(x) + log(5^(1/3)/10)
1 - log5 = log(x * (5^(1/3)/10))
Step 7: Simplify the right side of the equation.
Simplifying the expression on the right side of the equation:
1 - log5 = log(x * (5^(1/3)/10))
1 - log5 = log(x * (1/10)^(1/3))
Step 8: Apply the logarithmic property to convert the equation into exponential form again.
Rewriting the equation using the property log(a) = b is equivalent to a = 10^b:
1 - log5 = log(x * (1/10)^(1/3))
1 - log5 = x * (1/10)^(1/3)
Step 9: Solve for x.
To isolate x, we subtract 1 from both sides of the equation:
-x = (1/10)^(1/3) - log5
Finally, we solve for x by multiplying both sides of the equation by -1:
x = -((1/10)^(1/3) - log5)
So, the solution for x is x = -((1/10)^(1/3) - log5).