Solve each exponential equation. Give answers as decimals accurate to the thousands place.
1) log(base5)14=2x-9
2) log(base7)5=7-6x
3) (1/3)^(3x+5)=(9)^(8-x)
4) 4^(x-5)=8^(x+2)
so did you try?
i tried 3 and 4 but im not sure if its correct
3) -21
4) -16
and im not sure how to do 1 and 2
1) for example:
log5 (14) = 2x-9
5^[log5 (14)] = 14 = 5^(2x-9)
so the question is
5 to the what = 14 ?
say 5^z = 14 (we know that z is (2x-9))
z log 5 = log 14 using base 10 logs (or e or whatever, I use base 10)
z = log14/log5 = 1.146 / .699 = 1.64
so
2x-9 = 1.64
2x = 10.64
x = 5.32
try 3)
(1/3)^(3x+5)=(9)^(8-x)
(3x+5) log .333333333 = (8-x) log 9
(3x+5)(-.4771) = (8-x) (.9542)
(3x+5) = (x-8) 2 that was lucky :)
3 x + 5 = 2x-16
x = -21 agree
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or this way
(1/3)^(3x+5)=(9)^(8-x)
1^(3x+5)/3^(3x+5) = 3^(16-2x)
1 = 3^(x+21)
x+21 = 0
x = -21
4) 4^(x-5)=8^(x+2)
(2^2)^(x-5) = (2^3)^(x+2)
2^(2x-10) = 2^(3x+6)
2x-10 = 3x+6
x = -16 agree
2) log(base7)5=7-6x
5 = 7^(7-6x) = 7^z
log 5 = z log 7
z = .827
7 - 6 x = .827
6 x = 6.1729
x = 1.0288
or, for #1, just change the base to one you know
log_5(14) = log14/log5 = 1.64
now you are already at
2x-9 = 1.64
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Sure! To solve each exponential equation, we will use logarithms to eliminate the exponent and solve for the unknown variable. Let's go through each equation step by step:
1) log(base5)14 = 2x - 9
To solve for x, we need to isolate the variable. First, we can rewrite the equation using the definition of logarithm as:
5^(2x-9) = 14
Now we can solve for x by taking logarithm (base 5) on both sides:
log(base5)(5^(2x-9)) = log(base5)14
2x - 9 = log(base5)14
Now, we can isolate x by moving the constant to the other side of the equation:
2x = 9 + log(base5)14
Finally, divide by 2 to solve for x:
x = (9 + log(base5)14)/2
To obtain the value as a decimal, you can use a calculator or log tables to evaluate log(base5)14 and perform the necessary calculations.
2) log(base7)5 = 7 - 6x
Similarly, let's rewrite the equation using the definition of logarithm:
7^(7 - 6x) = 5
Now, take logarithm (base 7) on both sides:
log(base7)(7^(7 - 6x)) = log(base7)5
7 - 6x = log(base7)5
Move the constant to the other side:
6x = 7 - log(base7)5
Divide by 6 to solve for x:
x = (7 - log(base7)5)/6
Again, you can use a calculator or log tables to evaluate the logarithm and perform the necessary calculations to obtain the decimal value.
3) (1/3)^(3x+5) = (9)^(8 - x)
In this equation, we can rewrite both sides using the same base:
(3^(-1))^(3x+5) = (3^2)^(8 - x)
Applying the laws of exponents:
3^((-1)*(3x+5)) = 3^(2*(8 - x))
Now, we can equate the exponents:
-1(3x+5) = 2(8 - x)
Simplify:
-3x - 5 = 16 - 2x
Move all the terms containing x to one side:
-3x + 2x = 16 + 5
Simplify:
-x = 21
Finally, multiply both sides by -1 to solve for x:
x = -21
4) 4^(x-5) = 8^(x+2)
Since both 4 and 8 can be written as powers of 2, let's rewrite the equation accordingly:
(2^2)^(x-5) = (2^3)^(x+2)
Apply the laws of exponents:
2^(2*(x-5)) = 2^(3*(x+2))
Now, we can equate the exponents:
2(x-5) = 3(x+2)
Simplify:
2x - 10 = 3x + 6
Move the constant terms to the other side:
2x - 3x = 6 + 10
Simplify:
-x = 16
Multiply by -1 to solve for x:
x = -16
These are the solutions to each exponential equation.