Short answer is no. For details see the attached reference suggestion. Best wishes David Booth

Dan Li If you try to fit the curve the linear relationships assumptions will be violated. To predict more accurately a linear relationship is a practical approach that includes a constant rate of change of Y with X. Equal variance of residual is also an assumption to minimize the sum square error.

In a GLM you have:

1) a linear predictor function

2) a link function linking the linear predictor to the expectation and

3) a probability model of wich you want to fit the expectation depending the predictors.

The combination of 1 and 2 defines how the expectation is modelled. If you use a log-link function and the linear predictor b0 + b1x, then this relationship is log(E(Y|X)) = b0+b1x, so E(Y|X) = exp(b0)*exp(X)^b1, what is an exponential relationship between X and Y.

3 defines how the expectation and the variance are connected. The Gaussian model has an independent variance parameter, so in this case there is no relationship (the variance is constant). Other distributionmodels have particular relationships. When µ denotes the expectation, then the variance of a Bernoulli model is µ(1-µ), the varianse of a Poisson model is µ, that of a negative binomial model is µ² etc. Quasi-distribution models relax these relationships and allow to model mean-variance relationships that are some multiple of these functions.

So if the relationship between your predictor and the response is exponential, you may consider the log link-function. If the (residual or conditional) variance is constant, you would use a Gaussian model (with log-link). If it is not constant, you could use a Gamma model (with log-link), for which the variance is a multiple of µ (so it increases with increasing µ). If the variance increases stronger with increasing µ, you can use a quasi-modle with log-link and a better suited variance function.

You may also have a look into https://schmettow.github.io/New_Stats/glm.html