Transforming Exponential & Logarithmic Graphs

You’ll be able to

Locate the asymptote of a transformed exponential or logarithmic graph
State the domain and range of a transformed exponential or logarithmic function
Explain why horizontal transformations of a logarithm can be rewritten as vertical shifts

The asymptote moves with the graph

y = bˣ has a horizontal asymptote at y = 0, and y = log_b(x) has a vertical asymptote at x = 0. Under transformation the asymptote travels with the curve: a vertical shift up by k moves the exponential's asymptote to y = k, and a horizontal shift right by h moves the logarithm's asymptote to x = h. Finding the transformed asymptote first is the fastest route to the domain and range.

Transformed asymptotes, domains and ranges

y = a·b^(x−h) + k → asymptote y = k, domain all reals, range y > k (if a > 0) · y = a·log_b(x − h) + k → asymptote x = h, domain x > h, range all reals

For a < 0 the exponential's range flips to y < k. The reflection is in the sign of a, not in k.

Worked example

For f(x) = 3·2^(x−1) − 5, find the asymptote, domain, range, and y-intercept.

1.The vertical shift is −5, so the horizontal asymptote is y = −5.
2.Exponentials accept every real input, so the domain is all real numbers.
3.2^(x−1) is always positive and 3 is positive, so 3·2^(x−1) > 0, making f(x) > −5. Range: y > −5.
4.y-intercept: f(0) = 3·2^(−1) − 5 = 1.5 − 5 = −3.5.

Answer: Asymptote y = −5, domain all reals, range y > −5, y-intercept (0, −3.5). The graph rises from just above y = −5 on the left toward +∞ on the right, never reaching the asymptote.

Checkpoint

What is the domain of g(x) = log₃(2x − 6) + 4?

Horizontal stretches of a logarithm are vertical shifts

Here is something peculiar to logarithms. log(2x) = log 2 + log x, so horizontally compressing log x by a factor of 2 is identical to shifting it up by the constant log 2. No other elementary function does this. It is a direct consequence of the product rule, and it means a transformed logarithm often has several equally correct descriptions — worth knowing when your answer disagrees with a key that is nonetheless also right.

The log product rule as a transformation identity

log_b(cx) = log_b(c) + log_b(x) — a horizontal scaling equals a vertical shift

Similarly log_b(xⁿ) = n·log_b(x): raising the input to a power is a vertical stretch.

Worked example

Describe two different transformation sequences that both take y = log₂(x) to y = log₂(8x).

1.Reading it as written: the input is multiplied by 8, which is a horizontal compression by a factor of 8.
2.Applying the product rule: log₂(8x) = log₂ 8 + log₂ x = 3 + log₂ x.
3.That is a vertical shift up by 3.
4.Both descriptions produce exactly the same graph — the asymptote stays at x = 0 in either reading, and the point (1, 0) moves to (1, 3) in both.

Answer: Either a horizontal compression by a factor of 8, or a vertical shift up by 3 units. They are the same transformation because log₂ 8 = 3, an equivalence unique to logarithmic functions.

Tip

Track a single point through the transformation to check your work. y = log_b x always passes through (1, 0), and y = bˣ always passes through (0, 1). Following that one anchor point catches sign and direction errors immediately.

Reflections and the order of operations

Watch the difference between −log(x) and log(−x). The first reflects across the x-axis — the outputs change sign, and the domain stays x > 0. The second reflects across the y-axis — the inputs change sign, and the domain becomes x < 0. Whether the negative sits inside or outside the function is the whole question, and it decides the domain.

Checkpoint

What is the range of h(x) = −2·5^x + 7?

Answer the 2 checkpoints as you read.