Simplify.
1. (1/log base 3 of 60)+ (1/log base 4 of 60)+ (1/log base 5 of 60)
2. (log base 16 of x)+ ( log base 4 of x) + (log base 2 of x)=7
1/log_a(b) = log_b(a), so what you have is
log_60(3) + log_60(4) + log_60(5)
= log_60(3*4*5)
= log_60(60)
= 1
since 16=2^4=4^2, you have
log_16(x) + 2log_16(x) + 4log_16(x) = 7
7log_16(x) = 7
log_16(x) = 1
x=16
-3 · -u · -7v + 8v - 6uv
To simplify the expressions, we'll use the logarithmic identities and properties.
1. (1/log base 3 of 60)+ (1/log base 4 of 60)+ (1/log base 5 of 60)
Let's start by finding the common denominator of the three logarithms. The lowest common multiple of 3, 4, and 5 is 60. Therefore, we'll convert each logarithm to base 60 before summing them up:
(1/log base 3 of 60)= (1/log base 60 of 60)/ (1/log base 60 of 3)
(1/log base 4 of 60) = (1/log base 60 of 60)/ (1/log base 60 of 4)
(1/log base 5 of 60) = (1/log base 60 of 60)/ (1/log base 60 of 5)
Now, we'll add up the three fractions:
(1/log base 60 of 60)/ (1/log base 60 of 3) + (1/log base 60 of 60)/ (1/log base 60 of 4) + (1/log base 60 of 60)/ (1/log base 60 of 5)
Next, we can combine the fractions into a single fraction by finding the least common denominator:
[ (1/log base 60 of 60) * (1/log base 60 of 4) * (1/log base 60 of 5) + (1/log base 60 of 60) * (1/log base 60 of 3) * (1/log base 60 of 5) + (1/log base 60 of 60) * (1/log base 60 of 3) * (1/log base 60 of 4) ] / [ (1/log base 60 of 3) * (1/log base 60 of 4) * (1/log base 60 of 5) ]
Simplifying the expression further is difficult without precise values for the logarithms. So, the expression can be left in this form.
2. (log base 16 of x) + (log base 4 of x) + (log base 2 of x) = 7
We can rewrite the equation using the logarithm properties:
(log base 16 of x) + (log base 4 of x) + (log base 2 of x) = log base 16 of x + log base 2 of (x^2) = 7
Using the logarithm property log base b of a + log base b of c = log base b of (a * c), we have:
log base 16 of (x * x^2) = 7
Combining the terms under the same base:
log base 16 of (x^3) = 7
Since the base is 16, we can rewrite the expression in exponential form:
16^7 = x^3
Simplifying further:
x^3 = 16^7
Taking the cube root of both sides:
x = (16^7)^(1/3)
x = 16^(7/3)
The final simplified answer is x = 16^(7/3).
To simplify the given expressions, we need to rewrite the logarithmic terms using the properties of logarithms.
1. (1/log₃60) + (1/log₄60) + (1/log₅60)
We can start by using the property that states logₐb = 1/log_ba. This allows us to flip the base and the argument of the logarithm.
Therefore, we can rewrite the expression as:
(1/(log₆₀₃)) + (1/(log₆₀₄)) + (1/(log₆₀₅))
Now, we notice that the log base 60 is the same for all three terms. So, we can factor out the common denominator:
1/(log₆₀)
Now, the expression simplifies to:
1/log₆₀ = 1/(log₃60/log₃6₀) = 1/(log₃60/(2log₃3)) = 1/(log₃60/log₃3²) = 1/(log₃60/2log₃3)
Next, we can simplify further by multiplying the numerator and denominator by log₃3:
1/(log₃60/2log₃3) = 1/(log₃60/2) = 2/(log₃60)
Therefore, the simplified expression is 2/(log₃60).
2. (log₁₆x) + (log₄x) + (log₂x) = 7
In this case, we can use the property of logarithms that states logₐb + logₐc = logₐ(bc). This allows us to combine the logarithmic terms with the same base.
Using this property, we can rewrite the expression as:
log₁₆(x * x^2) = log₁₆(x^3)
Now, we can simplify further by converting the logarithmic equation into its exponential form. For logₐb = c, the exponential form is a^c = b.
Therefore, we have:
16^7 = x^3
Simplifying, we get:
x^3 = 2^7
x^3 = 128
Taking the cube root of both sides, we find:
x = 4
Therefore, the value of x that satisfies the equation is 4.