Does this graph follow a exponential distribution or a log-normal? 
command for the graph:
hist(c1,freq=FALSE)
lines(density(c1,adjust=2),col="darkblue",lwd=2)

I generated this sequence in R:
set.seed(106)
lambda1<-1/98
c1<-rexp(n=1000, rate=lambda1)

and did a z.test with a significance value of 5%:
z.test(c1,mu=98, alternative = "two.sided", sigma.x = sd(c1), conf.level = 0.95)

this is the output:
data:  c1
z = 0.51946, p-value = 0.6034
alternative hypothesis: true mean is not equal to 98
95 percent confidence interval:
  93.43883 105.85080
sample estimates:
mean of x 
 99.64482 

What I need to find out is if indeed the values follow a exponential distribution but I'm not sure.
 A: You defined c1 as a sample from an exponentially distributed population (in the line c1<-rexp(n=1000, rate=lambda1)), so that population must indeed be exponentially distributed.
(A sample itself can't be exponentially distributed; the only distribution a sample can be said to have is its empirical distribution, which must be discrete.)
A: I've got a dataset with similar data "waiting time in seconds", which belongs to an exponential distribution according to graphs. I have tried normal, lognormal as well, but it fits best an exponential distribution.
    library(gsheet)
    patience.data<-gsheet2tbl('https://docs.google.com/spreadsheets/d/1YIKOiA_xsg1ClJSYy0oZ0XYIxQzZfOqCEwolUeob_AU/edit?usp=sharing')

    wtime<-patience.data$sec
    hist(wtime,freq=FALSE)
    lines(density(wtime),col="red",lwd=2)
    #compare it with a theoretical normal distribution curve
    curve(dnorm(x,mean=mean(wtime),sd=sd(wtime)),
    add=TRUE, col="blue", lwd=2)
    legend("topright",col=c("blue","red"),legend =c("estimated normal density curve","kernel density curve"),lwd=2, bty = "n")


Let's fit data to an exponential distribution to the data and check it graphically
    require(fitdistrplus)
    fit.exp <- fitdist(wtime, "exp")
    plot(fit.exp)


The second and third graph look convincing
Let's fit now the histogram, density curve and exponential curve together
    fit.exp#get the estimated rate: 0.03482814
    hist(wtime,probability = TRUE)
    lines(density(wtime),col="red",lwd=2)
    curve(dexp(x, rate = 0.034828136), col = 3, lty = 2,lwd=2,
  add=TRUE)
    legend("topright",col=c("green","red"),
    legend =c("estimated exponential density",
             "kernel density"),
    lwd=2, bty = "n")


