Find the exact solution and a two-decimal-place approximation for it by using the Laws of Logarithms and the Change of Base Formula.
4^(5 − x) = 6
(a) using the Laws of Logarithms
exact solution
approximate solution
(b) using the Change of Base Formula
exact solution
approximate solution
4^(5 − x) = 6
(5-x) log4 = log6
5-x = log6/log4
x = 5 - log6/log4
5-x = log46
x = 5 - log46
(a) Using the Laws of Logarithms:
To find the exact solution, we can take the logarithm of both sides of the equation:
log base 4 of (4^(5 - x)) = log base 4 of 6
Using the Power Rule of logarithms, we can bring down the exponent:
(5 - x) log base 4 of 4 = log base 4 of 6
Since log base 4 of 4 is equal to 1, we can simplify further:
5 - x = log base 4 of 6
Now we can isolate x by subtracting 5 from both sides:
x = 5 - log base 4 of 6
For the approximate solution, we can evaluate the expression using a calculator:
x ≈ 5 - 0.9183 (rounded to two decimal places)
x ≈ 4.0817
(b) Using the Change of Base Formula:
To find the exact solution, we can rewrite the equation using the Change of Base Formula:
log base 4 of (4^(5 - x)) = log base 10 of (6) / log base 10 of (4)
Using the Power Rule of logarithms, we can bring down the exponent:
(5 - x) log base 4 of 4 = log base 10 of 6 / log base 10 of 4
Since log base 4 of 4 is equal to 1, we can simplify further:
5 - x = log base 10 of 6 / log base 10 of 4
Now we can isolate x by subtracting 5 from both sides:
x = 5 - (log base 10 of 6 / log base 10 of 4)
For the approximate solution, we can evaluate the expression using a calculator:
x ≈ 5 - (0.7782 / 0.6021) (rounded to two decimal places)
x ≈ 3.6797
So, the exact solution using the Laws of Logarithms is x = 5 - log base 4 of 6, and the approximate solution is x ≈ 4.0817.
The exact solution using the Change of Base Formula is x = 5 - (log base 10 of 6 / log base 10 of 4), and the approximate solution is x ≈ 3.6797.
(a) Using the Laws of Logarithms:
To solve the equation 4^(5 - x) = 6, we can take the logarithm of both sides of the equation:
log(base 4) (4^(5 - x)) = log(base 4) 6
Applying the Power Rule for logarithms, we can bring down the exponent (5 - x) as a coefficient:
(5 - x) * log(base 4) 4 = log(base 4) 6
Since log(base 4) 4 is equal to 1, we simplify:
5 - x = log(base 4) 6
To isolate x, we subtract 5 from both sides:
-x = log(base 4) 6 - 5
Finally, multiplying both sides by -1 gives:
x = 5 - log(base 4) 6
So the exact solution using the Laws of Logarithms is x = 5 - log(base 4) 6.
To find an approximate solution, we can calculate the value of log(base 4) 6 using a calculator and then substitute it into the equation.
(b) Using the Change of Base Formula:
The Change of Base Formula states that log(base a) x = log(base b) x / log(base b) a, where a, b, and x are positive real numbers with a, b ≠ 1.
To solve the equation 4^(5 - x) = 6, we can take the logarithm of both sides using any base, preferably a base that is easy to calculate:
log(base 10) (4^(5 - x)) = log(base 10) 6
Applying the Change of Base Formula with base 10:
(5 - x) * log(base 10) 4 = log(base 10) 6
Dividing both sides by log(base 10) 4:
5 - x = log(base 10) 6 / log(base 10) 4
To isolate x, we subtract 5 from both sides:
-x = log(base 10) 6 / log(base 10) 4 - 5
Finally, multiplying both sides by -1 gives:
x = 5 - log(base 10) 6 / log(base 10) 4
So the exact solution using the Change of Base Formula is x = 5 - log(base 10) 6 / log(base 10) 4.
To find an approximate solution, we can calculate the values of log(base 10) 6 and log(base 10) 4 using a calculator, then substitute them into the equation.
(a) Using the Laws of Logarithms:
Step 1: Take the logarithm of both sides of the equation using any base.
log(4^(5 - x)) = log(6)
Step 2: Apply the Power Rule of logarithms to simplify the left side of the equation.
(5 - x) log(4) = log(6)
Step 3: Solve for x by isolating it on one side of the equation.
5 - x = log(6) / log(4)
Step 4: Simplify the right side of the equation by using a calculator to evaluate the logarithms.
x = 5 - (log(6) / log(4))
This is the exact solution using the Laws of Logarithms.
To obtain the approximate solution, substitute the values of log(6) and log(4) into the equation and calculate the result.
(b) Using the Change of Base Formula:
Step 1: Apply the Change of Base Formula and rewrite the equation with a different base for the logarithm, such as base 10.
log(4^(5 - x)) = log(6)
(5 - x) log(base 10)(4) = log(base 10)(6)
Step 2: Solve for x by isolating it on one side of the equation.
5 - x = log(base 10)(6) / log(base 10)(4)
Step 3: Simplify the right side of the equation by using a calculator to evaluate the logarithms.
x = 5 - (log(base 10)(6) / log(base 10)(4))
This is the exact solution using the Change of Base Formula.
To obtain the approximate solution, substitute the values of log(base 10)(6) and log(base 10)(4) into the equation and calculate the result.