Can someone solve this and explain it please
4^log2 (2log2 5)
unsure how to explain how to solve it but i calculated it in my ti-84+ calculator with an equation solver that i downloaded.
4^log [2log (5)] = 21.56540031
2 2
im sorry i looked back at what i tried to write out and it turned out wrong, the two 2's are meant to be right after the g's in both the logs and as a subscript because it is log base 2.
To solve the expression 4^(log2(2log2(5))), let's break it down step-by-step:
Step 1: Simplify the innermost logarithm
Start by evaluating the logarithm inside the parentheses:
log2(5)
This represents the logarithm of 5 with base 2.
Step 2: Simplify the outer logarithm
Next, evaluate the logarithm outside the parentheses:
log2(2log2(5))
In this case, the base 2 logarithm is applied to the result of Step 1, which is log2(5).
Step 3: Simplify the exponent
After simplifying the logarithms, we can rewrite the expression as:
4^(log2(2log2(5))) = 4^(log2(2 * log2(5)))
In this case, the exponent is log2(2 * log2(5)).
Step 4: Apply the properties of logarithms
Using logarithmic identity log_a(bc) = c * log_a(b), we can simplify the exponent further:
log2(2 * log2(5)) = log2(2) + log2(log2(5))
The expression now becomes:
4^(log2(2) + log2(log2(5)))
Step 5: Evaluate the logarithmic terms
The first term, log2(2), simplifies to 1 since the logarithm of any number to its own base equals 1.
4^(1 + log2(log2(5)))
Step 6: Simplify the exponent
Combine the terms inside the exponent:
4^(1 + log2(log2(5))) = 4 * 4^(log2(log2(5)))
Step 7: Simplify further
We can rewrite the expression within the exponent as:
log2(log2(5)) = log2(log(5) / log(2))
So the expression now becomes:
4 * 4^(log2(log(5) / log(2)))
Step 8: Evaluate the logarithms
Logarithmic identity log_a(b/c) = log_a(b) - log_a(c) allows us to further simplify the exponent:
log2(log(5) / log(2)) = log2(log(5)) - log2(log(2))
Replacing the expression within the exponent, we now have:
4 * 4^(log2(log(5)) - log2(log(2)))
Step 9: Simplify the exponent
The expression within the exponent can be simplified further:
4^(log2(log(5)) - log2(log(2))) = 4^(log2(log(5))) / 4^(log2(log(2)))
Now, we have two exponential terms with the same base of 4.
Step 10: Evaluate the exponential terms
Evaluate each exponential term separately:
4^(log2(log(5))) and 4^(log2(log(2)))
To evaluate them, we can convert them into exponential forms with base 2:
2^(2log2(log(5))) and 2^(2log2(log(2)))
We're using the property that 4 is equal to 2^2.
Step 11: Simplify further
By applying the exponential property (a^(b*c) = (a^b)^c), we can simplify the expressions:
2^(2log2(log(5))) = (2^log2(log(5)))^2
2^(2log2(log(2))) = (2^log2(log(2)))^2
Step 12: Simplify the logarithms further
We can simplify the logarithmic expressions to:
2^log2(log(5))^2 and 2^log2(log(2))^2
Since the base of the logarithm matches the base of the exponential term, the logarithms cancel out:
log2(log(5))^2 and log2(log(2))^2
Step 13: Evaluate the remaining expression
Now we have:
4 * (log2(log(5))^2)/(log2(log(2))^2)
This is the simplified form of the original expression.
Please note that this expression cannot be further simplified without specific values for log(5) and log(2).