evaluate without using a calculator
[log(5)4 x log(2)10 ]/ [log(25)√10]
Oh boy, this looks like a job for Clown Bot, the calculator-less clown to the rescue! Hang on tight, we're going on a math adventure!
Okay, let's break it down step by step.
First, let's tackle the numerator: [log(5)4 x log(2)10].
To make things simpler, let's rewrite the logs with a common base. Let's use log base 10, because it's a number we're familiar with. So, we have:
[log(10)4 x log(10)10] / [log(25)√10].
Now, log(10)10 is equal to 1, because any number raised to the power of 1 is itself. Therefore, the numerator becomes:
[log(10)4 x 1] / [log(25)√10].
Simplifying further, we get:
[log(10)4] / [log(25)√10].
Now, let's deal with the denominator, [log(25)√10].
Using log properties, we can rewrite this as:
[log(25)10^(1/2)].
And we know that log(25) is equal to 2, because 25 is 5 squared. So, the denominator becomes:
[2 x 10^(1/2)].
Finally, we can substitute both the numerator and denominator into the original expression:
[log(10)4] / [2 x 10^(1/2)].
And there you have it! A simplified expression without using a calculator. However, if you'd like a more precise answer, I'm afraid we'll need to bring in a calculator-wielding expert. I did my best with a splash of humor, but sometimes, numbers just can't be clowned around with!
To evaluate the expression [log(5)4 x log(2)10] / [log(25)√10] without using a calculator, we need to simplify the expression using logarithmic properties.
First, let's simplify the numerator [log(5)4 x log(2)10]:
Using the logarithmic property log(a)b = log(c)b/log(c)a, we can rewrite log(5)4 as log(10)4 / log(10)5. Similarly, log(2)10 can be rewritten as log(10)10 / log(10)2.
Now we have [log(10)4 / log(10)5] x [log(10)10 / log(10)2].
Using another logarithmic property log(a)b x log(a)c = log(a)b/c, we can simplify the numerator further:
log(10)4 / log(10)5 = log(10)(4/5).
Now we have [log(10)(4/5)] x [log(10)10 / log(10)2].
Next, let's simplify the denominator log(25)√10:
Using the logarithmic property log(a)b = log(c)b/log(c)a, we can rewrite log(25)√10 as log(10)√10 / log(10)25.
Since √10 can be expressed as 10^(1/2), we can rewrite it further as log(10)10^(1/2) / log(10)25.
Simplifying this expression, we get log(10)10^(1/2) = 1/2.
Now, the denominator becomes 1/2 / log(10)25 = 2 / log(10)25.
Now we have [log(10)(4/5)] x [log(10)10 / (2 / log(10)25)].
Using the logarithmic property log(a)b / c = log(a)b - log(a)c, we can rewrite the expression as [log(10)(4/5)] x [log(10)10 - log(10)25/2].
log(10)10 equals 1, so the expression becomes [log(10)(4/5)] x (1 - log(10)25/2).
Finally, we can now evaluate [log(10)(4/5)] using the logarithmic property log(a)b = log(c)b/log(c)a:
log(10)(4/5) = log(e)(4/5) / log(e)10.
Using common logarithms (log base 10), we have log(10)(4/5) = log(4/5)/log(10).
Calculating log(4/5) and log(10), we can substitute the values and simplify the expression.
Thus, the evaluation of [log(5)4 x log(2)10] / [log(25)√10] without using a calculator requires several steps of simplification and substitution using logarithmic properties.
To evaluate the given expression without using a calculator, we can simplify it step by step.
Step 1: Recall the logarithmic identities:
- log(a, b) = log(c, b) / log(c, a)
- log(a, b^p) = p * log(a, b)
Step 2: Apply the first logarithmic identity:
[ log(5)4 x log(2)10 ] / [ log(25)√10 ]
= [ log(10)4 / log(10)5 ] x [ log(10)10 / log(10)2 ]
Step 3: Simplify the logarithms using the second logarithmic identity:
= [ log(10)4 / log(10)5 ] x [ 1 / log(10)2 ]
Step 4: Calculate each logarithm:
- log(10)4 is the exponent to which 10 must be raised to give 4. Since 10^2 = 100, we know that log(10)4 = 2.
- log(10)5 is the exponent to which 10 must be raised to give 5. Since 10^0.69897 ≈ 5, we know that log(10)5 ≈ 0.69897.
- Similarly, log(10)2 is the exponent to which 10 must be raised to give 2. Since 10^0.30103 ≈ 2, we know that log(10)2 ≈ 0.30103.
Step 5: Substitute the calculated values back into the expression:
= [ 2 / 0.69897 ] x [ 1 / 0.30103 ]
= (2 * 1) / (0.69897 * 0.30103)
= 2.862066
Therefore, the evaluated value of the given expression is approximately 2.862066.
converting to base 10 logs, we have
(log4/log5 * 1/log2)/(log √10 / log25)
= (log4/(log5 log2)) * 2log5 / (1/2)
= log4 log10 log25 / log2 log5 log√10
= (4 log2 log5) / (1/2 log2 log5)
= 8