Regression model when the dependent and independent variables show exponential distribution As the Title suggests i am trying to figure out what would be the regression model to use when both the dependent and independent variables show an exponential distribution.
Do I have to perform a log transform for both x and y and then use linear regression? below image shows the distribution of my dependent variable.

And below is the Bi variate analysis for x and Y

 A: I want to expand on WHuber's comment, as it's absolutely crucial to understanding regression.
Regression posits that you have some observations, $y$, that have variability that you think you can partially (or mostly) explain by considering the fact that the observations have some other characteristics in variables other than $y$. These are your predictors. For instance, there would be a lot of variability in human height, but there would be less variability if we accounted for age and gender. If you only know the distribution of human heights and have to guess the height of the next human you're going to see, you won't have much confidence in your guess. If, however, you know that the next human you see is going to be a 40-year-old woman instead of a 4-month-old boy, you will have more confidence in your guess.
You are making your prediction of the age conditioned on knowing tha age and gender. Everything in OLS regression about how the error term has constant variance, etc, refers to the conditional distribution, not the distributions of the predictive and response variables themselves.
You do not appear to want to do something like that.
You probably want to examine the copula between your two marginal exponential distributions. Remember how we say that variables $X$ and $Y$ are independent if and only if the joint distribution equals the product of the marginal distributions: $f_{X,Y}(x,y) = f_X(x)f_Y(y)$? Well what happens if we we tack on some other stuff so that $f_{X,Y}(x,y) = f_X(x)f_Y(y)C(x,y)$? When $C=1$, we have independence. When $C \ne 1$, there is some kind of dependence between the variables. Maybe it's simple correlation. Maybe it's something totally wild that causes the scatterplot to spell out your name. That dependence structure is governed by the copula between the two distributions.
A nice feature of the copula is that it doesn't care what the marginal distributions are. You can have normal marginals with the copula structure saying that the scatterplot should spell your name, and you could use that same copula structure. There's an R package to explore this.
# load the library and set a random seed for reproducibility
#
library(copula)
set.seed(2019)

# Define a "Gaussian" copula based on the 
# copula of the bivariate Gaussian distribution with rho=0.8
#
normal_copula <- normalCopula(param=0.8) 

# Define a "Gumbel" copula that has a tighter relationship when the 
# variables both take large values
#
gumbel_copula <- gumbelCopula(param=1.5)

# define a bivariate distribution with N(0,2) marginals and 
# normal_copula as the dependence structure
#
normal_normal_normal <- mvdc(normal_copula,c("norm","norm"),
list(list(mean=0,sd=2),list(mean=0,sd=2)))

# define a bivariate distribution with N(0,2) marginals and 
# gumbel_copula as the dependence structure
#
normal_normal_gumbel <- mvdc(gumbel_copula,c("norm","norm"),
list(list(mean=0,sd=2),list(mean=0,sd=2)))

# define a bivariate distribution with exp(1) marginals and 
# normal_copula as the dependence structure
#
exp_exp_normal <- mvdc(normal_copula,c("exp","exp"),list(1,1))

# define a bivariate distribution with exp(1) marginals and 
# gumbel_copula as the dependence structure
#
exp_exp_gumbel <- mvdc(gumbel_copula,c("exp","exp"),list(1,1))

# Draw 2500 observations from each of the four populations
#
NNN <- rMvdc(2500,normal_normal_normal)
NNG <- rMvdc(2500,normal_normal_gumbel)
EEN <- rMvdc(2500,exp_exp_normal)
EEG <- rMvdc(2500,exp_exp_gumbel)

# plot the scatterplots in a 2x2 array
#
par(mfrow=c(2,2))
plot(NNN,main="NNN")
plot(NNG,main="NNG")
plot(EEN,main="EEN")
plot(EEG,main="EEG")

When you plot those, you will notice that all of the plots look different. NNN and EEN, however, have the exact same kind of dependence structure, as to NNG and EEG! All that you're changing are the marginal distributions.
There are R packages out there to do copula estimation. We discussed kdecopula in school, but you may find one that you prefer.
(To clarify terminology, "copula" is akin to a CDF while "copula density" is akin to a PDF. In fact, they are CDFs and densities, respectively, but not of the original data.)
A: Regression assumes that the residuals will be normally distributed with constant variance. Sometimes non normal dependent variables can give normal residuals when fitted against some IV. But often, with right skewed data like this you can transform it by the natural log of your dependent variable; build your model away and then check the residuals. You can check a histogram of the residuals to see if it is normally distributed, alternatively you can check the QQ plot for a straight line.
To check constant variance you can plot the residuals against the fitted values, and look for any obvious pattern in the plot (usually a fan where variance increases for larger values).EDIT: if your residuals have constant variance, it will not have any pattern.
If that doesn't work, another approach is to try and use generalized linear models to fit it to distributions that can handle right skewed data such as the gamma, or the inverse Gaussian distributions. 
When you interpret the model equation you have to take the exponential and it's a multiplicative model. Well, basically use this page, because it explains it a lot better than i can
You don't have to log your independent variable. Regression makes no assumption about it. I mean it's possible that your dependent variable depending on your independent variable unlogged is normally distributed, but that's not normally the case.
