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# Analyzing graphs of exponential functions: negative initial value

Given the graph of an exponential function with a negative initial value, Sal finds the formula of the function and solves an equation.

## Want to join the conversation?

• Is there an algebraic way of determining the unknown value of an exponent? For example, 5^x = 625, how can we algebraically determine x? In the video, Sal wrote the values of 5^1, 5^2, 5^3, and 5^4 from memory. But how can the value of the exponent be calculated?
• I highly recommend watching the video made by Eddie Woo on YouTube called “Introduction to Logarithms (1 of 2: Definition)”
• I couldn’t understand why did 1/5^x=1/625 become 5^x=625? Please, help.
• I can always reciprocate both sides of an equation (if both are only fractions). Think about cross multiplying the fraction, you get 5^x * 1 = 625 * 1, and multiplying by 1 does not change a number, thus 5^x = 625.
• At what do you think the siren is?
• Probably just the cops showing up at his house to give him the "Greatest Math Teacher of All Time" award.
• Why do we derive the common ration by f(0) or f(1).. Why not f(0), f(1.5)?
• The common ratio is found by dividing one term by the term before it.
Using the formula, that means that we would divide one term ( ar^x) by the term before it ( ar^(x-1) ). to give us an answer of r.
If you used f(1.5) when wanting to find the common ratio,
you would need to divide by ar^.5 in order to still arrive at the answer of r.
To answer your question, it makes things much simpler to stick with whole numbers!
• How can we algebraically solve a problem like this: 5(to the x power) = 625
what is the mathematical way to find x ?
• logarithms are the inverse of exponents in the same way that division is the inverse of multiplication, given x^y, log_x x^y = y aka log base x of x to the power of y equals y, or x^log_x(y) = y aka x to the power of the log base x of y is y eg. 10^(log_10 10^3) = 10^3.

log_5 (5^x) = x
since we're saying that 5^x = 625 then
log_5 (625) = x

if log_5 (5^x) != log_5 (625) then 5^x != 625

A useful exponent/log/power rule is that log_n x = log_10 x / log_10 n, this is particularly useful since many calculators only provide log_10 not log_<whatever you want> (if it's labeled log then it's log_10 aka log base 10, ln is the "natural log" or log_e, in general it works for any log base eg. log_y x / log_y n = log_n x, so log_5 625 = log_10 / log 5 = ln 625 / ln 5 = ...)

a calculator will tell you that log_5 625 is 4.

Note that logarithms are not limited to whole numbers, you can take the log of 142 and get 2.15... since 10 to the power of 2.15... is 142.
• Is this graph growth or decay or neither? If a is negative and b is between 0 and 1, the graph is increasing, but hits the asymptote at zero. Does "a" have to be always positive to be classified as growth and decay ? Is there such a thing as negative growth and decay?
• Annabelle,

Even though the numbers are negative, this is a graph of exponential growth - as `x` increases `y` increases.
• Let's see if someone can help me, I have a tricky one. I have understood that the base of an exponential function can't be negative, since with fractional exponential values, you'd end up with complex numbers: f(x) = -3^x and you evaluate x = 1/2, you get f(1/2) = -3^(1/2), which leads to complex numbers

Now, then why when I tried f(x) = -3x in DESMOS graphing calculator I got a graph? How can this graph exist?

P.S. Sorry if there was a better lecture for where to post this question, this is the best I found.
• You made a common error in the way you entered the function.
Remember -3^2 and (-3)^2 are different.
-3^2 = -(3*3) = -9
(-3)^2 = (-3)(-3)

The base of the exponent in f(x)=-3^x is "3", not "-3". And the "-" in front takes the standard exponential function and reflects it across the x-axis. This is why you got a graph.

To have a base of -3, you need to enter the function as: f(x)=(-3)^x

Hope this helps.
• can this kind of graph curve into a parabola?
(1 vote)
• A parabola is the graph of a second-degree polynomial function,
𝑦 = 𝐴𝑥² + 𝐵𝑥 + 𝐶,
which can never be equivalent to an exponential function,
𝑦 = 𝑎𝑟^𝑥