use exponential form to evaluate log_8 2 show all steps
To evaluate log_8 2, we can convert it to exponential form.
Let's assume that log_8 2 = x.
According to the definition of logarithm, we have:
8^x = 2
To solve for x, we can rewrite 8 as a power of 2:
(2^3)^x = 2
Using the property of exponents, we can multiply the exponents:
2^(3x) = 2
Now, since the bases of both sides are the same, we can equate the exponents:
3x = 1
Dividing both sides by 3:
x = 1/3
Therefore, log_8 2 = 1/3.
To evaluate log_8 2 using exponential form, we need to express it as an equation in exponential form, where the base is 8, the exponent is unknown and the result is 2.
In exponential form, the equation would be: 8^x = 2
To solve this equation for x, we can take the logarithm with base 8 of both sides:
log_8 8^x = log_8 2
Since the logarithm and exponentiation are inverse operations, the left side simplifies to just x:
x = log_8 2
Now, let's evaluate log_8 2 using the change of base formula, which states that any logarithm with base b can be expressed in terms of logarithms with different bases. In this case, we will use the natural logarithm (ln) as the base, but you can also use common logarithm (log) or any other logarithm base:
x = log_8 2 = ln 2 / ln 8
Using a calculator, we can find that ln 2 ≈ 0.693 and ln 8 ≈ 2.079.
Plugging in these values, we get:
x ≈ 0.693 / 2.079 ≈ 0.333
Therefore, log_8 2 is approximately 0.333.
To evaluate the logarithm log_8 2 using exponential form, we need to rewrite it as an exponential equation.
In exponential form, the logarithm log_a b can be expressed as:
a^x = b
where a is the base of the logarithm, x is the exponent, and b is the value of the logarithm.
In this case, we have log_8 2.
So, we can rewrite it as:
8^x = 2
Now, our goal is to determine the value of x.
To solve this equation, we need to find the exponent that can be raised to 8 to get 2.
Let's convert both sides of the equation to common bases:
2 can be expressed as 8^(1/3) since 8^(1/3) = 2.
So, we have:
8^x = 8^(1/3)
To find the value of x, we can equate the exponents:
x = 1/3
Therefore, log_8 2 can be evaluated as x = 1/3.