Explain how to do these please.
1. solve x=log(base2)*1/8 by rewriting in exponential form.
2. Solve log(3x+1)=5
3. Solve log(base x)8=-1/2
x = log2 (1/2)^3
x = 3 log2 (1/2)
x/3 = log 2(1/2) = log 2 (1) - log2(2)
but log2 (1) = 0 and log 2 (2) = 1
so
x/3 = -1
x = -3
log(3x+1)=5
I get we mean log base 10
10^log(3x+1) = 3x+1
so
3x+1 = 10^5
3 x = 10^5 -1 = 9999
x = 3333
log(base x)8=-1/2
x^log(base x)8 = 8
so
8 = x^(-1/2) = 1/sqrt x
sqrt x = 1/8
x = 1/64
1. To solve the equation x = log(base2)(1/8) by rewriting it in exponential form, we need to remember the basic definition of logarithms. A logarithm is the inverse operation of exponentiation. The logarithmic form log(baseb)(a) = c tells us that b raised to the power of c equals a. In this case, we have x = log(base2)(1/8). To rewrite it in exponential form, we need to isolate the logarithm.
First, rearrange the equation as follows:
log(base2)(1/8) = x
Then, apply the definition of logarithms:
2^x = 1/8
Now, simplify the right side of the equation:
2^x = 1/2^3
Since 1/2^3 equals 2^(-3), we can rewrite the equation as:
2^x = 2^(-3)
To solve for x, we can equate the exponents:
x = -3
Therefore, the solution to the equation x = log(base2)(1/8), when rewritten in exponential form, is x = -3.
2. To solve the equation log(3x+1) = 5, we follow the same principles of logarithms. The logarithmic form log(baseb)(a) = c tells us that b raised to the power of c equals a. In this case, we have log(3x+1) = 5. To solve for x, we need to isolate the logarithm.
First, get rid of the logarithm by applying exponentiation. Raise both sides of the equation to the power of the base, which in this case is 10:
10^(log(3x+1)) = 10^5
The logarithm log(baseb)(b^a) = a property allows us to cancel out the logarithm, giving us:
3x + 1 = 10^5
Now, simplify the right side of the equation:
3x + 1 = 100,000
Next, isolate the variable x:
3x = 100,000 - 1
3x = 99,999
Finally, solve for x by dividing both sides of the equation by 3:
x = 33,333
Therefore, the solution to the equation log(3x+1) = 5 is x = 33,333.
3. To solve the equation log(base x)8 = -1/2, we apply the definition of logarithms. The logarithmic form log(baseb)(a) = c tells us that b raised to the power of c equals a. In this case, we have log(base x)8 = -1/2. To solve for x, we need to isolate the logarithm.
First, rewrite the equation using exponential form:
x^(-1/2) = 8
To eliminate the negative exponent, we can write the equation as a reciprocal:
1/sqrt(x) = 8
Now, cross-multiply to get rid of the fraction:
1 = 8 * sqrt(x)
Simplify the right side of the equation:
1 = 8√x
Next, square both sides of the equation to eliminate the square root:
1^2 = (8√x)^2
1 = 64x
Now, solve for x by dividing both sides of the equation by 64:
x = 1/64
Therefore, the solution to the equation log(base x)8 = -1/2 is x = 1/64.